Verificación de identidades trigonométricas

Las identidades trigonométricas pueden ser de formas variadas. Ya hemos visto distintos tipos de identidades estándar, como las fórmulas del ángulo doble, la suma de ángulos, la diferencia de ángulos, las identidades recíprocas, etc. Éstas son las identidades estándar que sirven de base a las demás identidades que se derivan de ellas. Estas identidades que pueden descomponerse en identidades fundamentales suelen denominarse identidades secundarias.

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Verificación de identidades trigonométricas Verificación de identidades trigonométricas

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    De hecho, se puede formar un gran número de identidades a partir de las identidades fundamentales. Para verificar dichas identidades, utilizamos el mismo tipo de procedimiento que empleamos para resolver distintas ecuaciones, es decir, resolvemos la identidad común.

    Reglas de verificación de las identidades trigonométricas

    1. Si hay una identidad formada por diferentes funciones trigonométricas, como una identidad formada por tangente y seno, intenta separarlas.

    2. Intenta que la tangente esté en un lado y el seno en el otro de la ecuación, así será más fácil aplicar las identidades fundamentales.

    3. Ahora lo más importante es saber qué identidad fundamental hay que aplicar.

    4. Convierte la ecuación en la misma función, es decir, por ejemplo, convierte todas las tangentes en senos o todos los senos en tangentes.

    5. Después de realizar todos los pasos anteriores, si el lado derecho (LD) de la ecuación es igual al lado izquierdo (LH), entonces la identidad es cierta: LHS=RHS

    ¿Cómo demostrar algebraicamente una identidad trigonométrica?

    Podemos demostrar las identidades trigonométricas algebraicamente resolviendo el LHS y el RHS por separado. Aquí primero resolvemos el lado complejo de la ecuación y comprobamos si tanto el LHS como el RHS son iguales o no. Veamos algunos ejemplos.

    Comprueba quecot2θ-cos2θ=cot2θ cos2θ es una identidad coherente.

    Solución:

    Paso 1: Sumando cos2θ{"x":[[229,226,222,218,213,207,201,196,192,188,185,183,181,181,181,181,181,184,188,191,196,201,207,212,217,222,228,233,237,242,246,250,253,255,256,256,256,256,256,255,253,251,248,246,244,242,241,239,238,237,237,236,236,236,236,236,236,237,238,239,241,243,246,248,252,255,259,263,268,272,275,279,282,284,286,287,287,287,287,287,284,282,280,277,275,272,269,267,264,262,260,260,259,259,259,259,260,263,267,272,278,285,292,299,307,313,319,325,329,332,335,337,338,339,339,339,339,338,337,335,332,331,328,327,325,324,323,322,322,322,322,323,325,327,330,332,333,335,335,335,335,334,330,327,324,321,318,315],[362,362,362,364,366,370,374,379,383,387,390,393,395,396,397,397,397,397,397,396,393,390,387,385,382,380,378,378,378,382,386,392,398,404,411,417,423,427,431,433,435],[418,416,416,414,413,412,411,410,410,410,410,410,411,413,416,419,423,427,431,435,440,443,447,450,452,452,453,453,453,453,452,450,447,445,441,438,434,430,426,423,420,417,416,415,415,415,416,420,426,434,443,452,461,469,476,481,484]],"y":[[281,281,282,286,290,294,299,304,308,313,318,322,326,330,333,336,338,340,341,342,342,342,342,341,339,336,333,330,327,324,321,317,314,310,307,304,302,299,297,295,294,293,293,293,293,293,293,294,296,299,303,307,310,315,318,322,326,329,332,335,338,339,341,341,341,341,341,341,337,334,330,326,321,317,312,307,303,298,295,293,291,290,289,289,289,289,292,294,298,301,304,307,310,312,315,317,319,320,322,322,322,322,322,319,316,313,310,306,303,300,296,293,290,287,286,284,283,283,282,282,282,282,283,284,286,288,291,293,296,299,303,306,309,312,315,317,319,322,324,327,329,331,334,336,337,338,338,338],[238,233,230,228,225,222,220,218,216,215,214,214,214,214,215,217,219,222,225,228,232,235,239,243,246,248,250,251,252,252,252,251,250,248,246,245,244,243,243,243,243],[291,294,298,303,308,314,320,325,329,333,336,339,340,341,342,342,342,341,337,333,329,324,318,313,307,302,297,292,288,284,282,281,279,279,279,279,280,283,286,289,293,297,300,304,307,310,312,314,315,315,315,315,315,315,315,315,315]],"t":[[0,94,108,115,124,134,140,151,157,168,174,184,190,201,207,217,223,234,240,251,257,268,273,284,291,301,307,317,323,334,340,350,357,368,373,384,390,401,407,418,423,434,440,451,457,468,474,485,490,501,507,518,523,534,540,550,557,567,574,584,590,601,607,617,624,634,640,650,657,667,674,684,690,700,707,717,724,734,740,750,757,767,774,784,790,801,807,818,824,834,840,849,857,867,874,884,890,901,907,917,924,934,940,950,957,967,974,984,990,1001,1007,1017,1024,1034,1040,1050,1057,1067,1074,1084,1090,1101,1107,1115,1124,1134,1140,1151,1157,1167,1174,1184,1190,1201,1207,1217,1224,1234,1240,1251,1257,1267,1274,1284,1290,1301,1307,1317],[1681,1702,1718,1724,1736,1741,1753,1757,1769,1774,1786,1791,1802,1808,1819,1824,1836,1841,1852,1857,1869,1874,1885,1891,1902,1907,1918,1924,1935,1966,1974,1984,1991,2002,2007,2018,2024,2035,2041,2051,2057],[2388,2453,2466,2474,2486,2491,2503,2508,2520,2524,2536,2541,2553,2558,2570,2574,2587,2591,2603,2608,2619,2624,2635,2641,2652,2658,2668,2674,2685,2691,2702,2709,2719,2724,2735,2741,2752,2758,2768,2774,2785,2791,2802,2808,2819,2824,2835,2841,2852,2858,2866,2874,2883,2891,2902,2908,2919]],"version":"2.0.0"} en ambos lados de la ecuación, obtenemos

    cot2θ=cos2θ+cos2θ cot2θ =cos2θ(1+cot2θ)

    Paso 2: Se observa que 1+cot2θ es una identidad fundamental, es decir

    cosec2θ=1+cot2θ

    Paso3 : Sustituyendo esta identidad, obtenemos

    cot2θ=cos2θ cosec2θ

    Paso 4: Ahora, utilizaremos una identidad recíproca para la cosecante,cosecθ=1sinθ{"x":[[159,158,157,156,154,153,151,149,146,143,141,138,137,136,137,139,142,145,148,152,156,160,164,168,170,173,174,175,175,175,174,173,173,173,173,173,174,175,177,180,183,187,189,192,195,196,197,198,198,197,196,194,191,189,186,184,182,181,181,181,183,187,192,197,204,210,216,221,225,229,232,234,234,234,234,233,232,231,229,228,227,227,227,228,229,231,232,233,233,231,229,227,225,224],[257,256,255,255,254,255,256,257,258,260,261,263,264,265,265,265,264,262,260,257,254,252,250,249,249,251,254,259,263,269,272],[294,292,290,289,288,286,285,283,281,280,279,280,283,286,291,295,299],[326,326,326,325,325,325,324,323,322,321,321,320,320,321,323,326,329,332,334,337,338,339,340,340,339,338,336,333,330,327,324,322,321,321,323,326,331,337,342],[407,403,405,408,413,418,422,425],[400,398,397,398,401,405,410,416,424,430,435,438],[544,544,544,544,543,541,540,539,538,538,538,537,537,537,537,537],[502,501,500,498,500,505,511,522,534,550,565,579,593,605,614,621,625,626,625,622,620],[499,500,501,501,502,503,504,504,505,505,505,504,502,500,497,494,492,490,490,490,491,493,497,499,502,503,504,503,502,499,496,493,490,487,486,485,485,485],[519,519,519,519,519,519,519,520,520,520,520,521,522,524,526,528,531],[531,530,527,527,528,529,531,534],[540,538,538,537,538,538,539,539,540,540,540,540,540,540,542,544,546,548,551,553,554,555,556,556,557,557,557,558,558,559,561,563,565],[582,582,582,582,582,583,583,583,583,582,582,582,582,582,583,584,586,588,591,593,595,597,599,600,600,600,599,597,595,592,590,587,585,584,584,584,585,588,591,596,601]],"y":[[157,155,154,153,152,151,151,151,152,155,159,164,169,174,177,180,181,182,182,182,180,178,175,171,168,164,161,158,156,155,155,157,160,163,166,170,172,175,178,179,179,179,178,176,173,170,167,164,160,157,155,153,152,152,154,156,158,161,165,168,171,173,174,175,175,173,170,167,163,159,155,152,149,146,145,144,144,144,145,148,151,154,158,162,165,169,171,173,175,175,175,174,173,171],[159,158,158,157,157,157,157,157,157,157,157,156,155,154,152,151,150,149,149,151,154,158,162,166,170,173,174,174,173,171,169],[149,148,147,147,148,149,152,155,159,164,169,172,175,177,178,178,178],[153,152,151,150,149,148,150,154,158,163,168,173,178,181,183,184,183,181,178,173,168,163,157,151,146,142,140,139,139,141,144,148,152,156,160,162,164,165,165],[148,148,148,147,147,146,146,146],[161,161,161,161,161,160,159,158,157,156,156,155],[111,107,106,108,111,116,122,128,134,139,145,149,152,154,155,156],[186,185,185,183,182,182,181,179,178,177,176,175,175,174,174,174,174,175,176,177,177],[232,230,230,229,228,226,224,223,221,220,219,217,217,216,217,219,221,224,227,230,232,235,237,240,241,243,243,244,244,244,244,244,243,242,241,239,238,237],[226,225,224,223,222,223,225,227,230,233,236,239,241,242,242,242,239],[206,206,205,204,204,205,206,207],[220,219,218,218,219,220,222,224,227,229,232,233,235,234,232,229,226,224,221,220,219,219,220,221,224,226,229,232,234,235,237,237,237],[221,220,219,218,217,217,218,220,222,225,229,232,235,238,239,240,240,239,237,233,230,226,221,217,212,208,205,204,203,203,205,208,212,216,220,223,226,228,229,229,228]],"t":[[0,8,12,20,27,36,44,52,63,69,79,86,96,102,112,119,129,136,146,152,162,169,179,191,196,209,212,219,229,236,253,269,279,286,296,302,313,319,329,336,346,352,363,369,379,386,396,403,413,419,430,436,446,453,466,469,479,491,496,503,513,519,529,536,546,553,563,569,580,586,596,603,613,619,629,636,646,654,663,669,679,686,694,703,712,719,730,736,746,763,769,779,790,796],[934,944,961,969,986,1011,1019,1030,1036,1046,1053,1061,1069,1080,1091,1097,1103,1112,1119,1130,1136,1146,1153,1163,1169,1180,1186,1197,1203,1213,1218],[1697,1704,1713,1720,1728,1736,1747,1753,1764,1770,1780,1786,1797,1803,1814,1820,1830],[2076,2080,2084,2086,2095,2103,2145,2153,2164,2170,2180,2186,2197,2203,2214,2220,2231,2237,2247,2253,2264,2270,2281,2287,2297,2303,2313,2320,2330,2336,2347,2353,2364,2370,2381,2387,2397,2403,2414],[2767,2772,2803,2812,2820,2831,2837,2844],[2943,2948,2954,2978,2987,2997,3003,3014,3020,3031,3037,3048],[3424,3429,3433,3462,3471,3480,3487,3499,3504,3517,3521,3531,3537,3548,3554,3564],[3832,3837,3841,3843,3862,3870,3881,3887,3898,3904,3915,3921,3931,3937,3948,3954,3965,3971,3981,3987,3995],[4355,4363,4368,4376,4376,4383,4392,4392,4404,4409,4409,4423,4431,4437,4448,4454,4465,4471,4482,4487,4498,4504,4515,4521,4531,4537,4548,4554,4565,4571,4582,4587,4598,4604,4615,4621,4631,4637],[4757,4762,4768,4784,4804,4829,4838,4848,4854,4865,4871,4882,4888,4899,4904,4915,4921],[5008,5013,5020,5024,5054,5063,5071,5082],[5223,5231,5236,5246,5279,5288,5298,5306,5316,5323,5332,5338,5349,5379,5388,5398,5404,5415,5421,5432,5438,5448,5454,5464,5471,5481,5488,5499,5504,5515,5521,5532,5538],[5718,5726,5730,5746,5765,5780,5788,5799,5805,5816,5821,5832,5838,5846,5855,5865,5871,5880,5888,5899,5908,5916,5925,5933,5938,5949,5955,5965,5971,5982,5988,5999,6005,6016,6021,6032,6038,6049,6055,6065,6071]],"version":"2.0.0"} y elevando al cuadrado ambos lados, obtenemos

    cosec2θ=1sin2θ{"x":[[105,104,103,102,101,100,98,97,94,92,90,88,87,86,88,90,93,97,102,107,113,118,123,127,130,132,134,134,134,134,133,133,132,132,131,131,131,132,132,133,134,136,138,140,142,144,146,147,148,148,148,148,146,144,142,140,138,137,135,135,135,135,137,140,144,149,155,161,168,173,177,181,183,185,185,184,183,182,182,182,182,182,183,184,184,184,183,181,179,175,173,171],[207,206,205,205,206,207,208,209,211,212,213,214,215,216,216,215,214,213,210,208,206,205,205,206,208,212,215,220,224],[252,250,249,248,247,246,243,241,239,238,238,238,240,242,246,250,254,258,260],[265,264,261,260,260,261,263,266,269,271,273,274,274,274,273,271,270,268,267,266,266,268,271,276,279],[307,306,305,305,304,304,302,302,301,301,301,302,303,305,307,310,312,315,317,319,321,322,322,322,321,319,317,314,312,310,308,307,307,307,308,310,313,318,323,325],[380,379,377,376,377,379,382,385,390,394,396],[377,374,373,375,378,382,388,394,400,404],[545,546,547,547,548,548,548,547,546,545,543,542,541,541,541,541],[506,504,501,502,506,513,521,532,547,564,580,597,614,629,642,655,665,672,678,679],[493,494,495,495,495,495,495,494,493,491,489,488,487,486,487,488,490,492,494,495,496,496,494,492,489,486,483,481,480,480,481],[513,513,513,513,513,513,513,512,512,512,512,512,513,514,515],[520,520],[532,532,532,533,533,534,534,534,534,534,534,534,533,533,534,536,539,542,545,547,549,550,550,550,550,550,550,551,552,553,554,556,559],[569,567,567,567,566,566,566,568,570,573,577,579,582,583,583,583,582,579,577,575,573,572,573,576,580,585],[617,619,620,620,621,621,621,621,621,620,620,619,619,619,619,620,622,624,627,630,634,637,640,642,644,645,646,645,643,641,638,635,631,627,624,623,622,623,626,630,635,641,645]],"y":[[231,231,230,230,230,230,230,231,234,237,242,248,253,258,261,263,264,264,264,263,260,258,255,252,249,246,242,240,239,238,238,239,242,245,249,252,255,259,262,264,265,266,266,264,262,259,255,252,248,244,240,237,235,234,234,234,236,237,240,243,247,250,254,256,258,258,258,255,252,249,246,243,240,238,236,237,238,240,243,246,249,253,256,259,261,263,264,264,264,263,262,261],[245,245,245,246,246,246,246,246,245,244,243,242,240,239,238,237,237,237,239,241,245,248,251,254,256,257,257,257,255],[232,230,230,230,231,232,235,239,243,246,249,252,254,254,254,253,252,249,247],[198,198,196,195,194,193,192,191,190,190,190,192,194,197,199,202,205,208,209,210,211,211,210,208,207],[227,226,223,224,227,231,235,240,245,250,255,259,261,263,264,264,262,260,257,252,247,242,235,229,224,220,218,217,217,218,220,224,227,231,234,237,239,240,241,241],[230,230,229,229,229,229,229,229,229,229,229],[245,244,244,244,244,244,244,243,242,242],[205,200,199,198,198,199,201,204,209,214,219,224,230,235,240,242],[272,272,271,271,271,271,270,269,268,267,266,265,264,264,264,264,264,264,265,265],[323,322,320,318,317,315,313,312,311,311,311,312,314,317,320,322,325,328,330,332,334,335,336,337,338,338,337,336,334,332,330],[316,315,314,313,314,315,317,320,322,325,328,331,333,335,335],[297,298],[315,313,312,313,314,316,318,320,323,326,328,330,332,333,331,328,324,321,318,315,314,314,316,318,321,324,327,330,332,335,336,337,337],[294,291,290,289,288,287,286,286,286,285,285,286,287,289,291,294,297,299,302,304,305,305,305,305,303,302],[321,317,316,315,314,313,312,313,317,321,325,329,334,338,343,346,348,349,350,348,346,343,338,333,327,320,313,306,301,299,297,297,299,301,305,309,312,316,318,321,323,324,325]],"t":[[0,10,17,35,41,52,58,66,75,85,91,101,108,118,125,135,141,152,158,168,175,185,191,202,208,218,225,235,242,251,275,284,292,302,308,318,325,335,342,352,358,369,375,385,392,402,408,418,425,435,442,452,458,468,475,485,492,502,508,518,525,535,542,550,558,568,575,585,592,601,608,618,625,635,642,675,684,692,702,708,718,725,735,742,752,758,769,775,785,792,802,807],[998,1008,1025,1067,1075,1085,1092,1102,1108,1119,1127,1136,1142,1154,1159,1169,1175,1186,1192,1203,1208,1219,1225,1236,1242,1253,1259,1269,1275],[1719,1728,1732,1739,1747,1756,1759,1769,1776,1786,1792,1803,1809,1819,1826,1836,1842,1853,1858],[2047,2055,2058,2062,2076,2086,2093,2103,2109,2120,2126,2136,2142,2153,2159,2169,2175,2186,2192,2203,2209,2220,2226,2236,2241],[2625,2634,2638,2659,2668,2676,2684,2693,2703,2709,2720,2726,2737,2742,2753,2759,2770,2776,2786,2792,2803,2809,2819,2826,2837,2842,2853,2859,2870,2876,2887,2893,2903,2909,2920,2926,2935,2942,2957,2960],[3315,3320,3327,3343,3359,3370,3378,3387,3393,3404,3408],[3553,3562,3569,3587,3593,3604,3609,3620,3626,3633],[4024,4029,4035,4043,4054,4060,4071,4076,4087,4093,4104,4110,4121,4126,4135,4142],[4456,4464,4469,4478,4489,4493,4504,4510,4521,4527,4538,4543,4554,4560,4571,4577,4588,4593,4604,4609],[5027,5038,5044,5052,5066,5071,5080,5088,5093,5104,5110,5121,5127,5138,5143,5155,5160,5171,5177,5188,5193,5205,5210,5221,5227,5238,5243,5255,5260,5271,5277],[5405,5412,5423,5438,5463,5471,5477,5488,5493,5505,5510,5521,5527,5537,5542],[5697,5710],[6195,6203,6208,6236,6244,6255,6261,6270,6277,6288,6294,6305,6311,6319,6360,6369,6377,6387,6394,6405,6411,6422,6427,6439,6444,6455,6461,6472,6477,6489,6494,6505,6511],[6729,6734,6738,6744,6754,6761,6777,6786,6794,6805,6811,6822,6827,6839,6844,6856,6861,6871,6877,6888,6894,6906,6922,6927,6939,6944],[7265,7270,7275,7282,7286,7294,7305,7337,7344,7356,7361,7373,7378,7389,7394,7406,7411,7423,7428,7439,7444,7456,7461,7472,7478,7486,7494,7505,7511,7523,7528,7539,7544,7555,7561,7572,7578,7589,7594,7605,7611,7622,7627]],"version":"2.0.0"}

    Paso 5: Sustituyendo la identidad anterior, obtenemos

    cot2θ=cos2θsin2θ

    Paso 6: Ya sabemos que cotθ=cosθsinθy elevándolo al cuadrado, se obtiene el paso anterior.

    Por tanto, LHS=RHS{"x":[[268,268,268,261,258,257,256,254,254,254,255,259,263,269,277,281,294,300,316,322,340,345,350,360,369,377,380,383,385,387],[427,427,426,426,426,427,428,428,430,430,431,432,432,433,434,434,434,435,435],[422,422,423,423,424,426,431,436,441,444,447,453,459,470,476,480,482,488,490,492,494,494,495,495,495,494,493,492,491,490,490,489,487,487,486,485,484,484,484,484,484,484,484,484,484,484],[601,606,608,608,609,609,608,607,606,598,591,583,573,564,556,553,543,541,539,540,541,545,550,556,559,562,566,575,578,586,587,589,588,586,581,575,572,565,557,556],[646,646,647,648,650,656,658,666,670,677,685,690,696,698,702,705],[617,618,620,621,626,633,640,644,651,658,661,668,673,674,675],[818,823,824,826,827,827,826,825,823,820,816,813,811,810,810,810,810,810,810,810,811,811],[819,819,819,820,824,830,833,844,848,859,866,871,873,876,877,880,880,879,877,870,864,857,850,844,841,837,836,836,836,837,839,844,848,855,861,867,871,877,886,891,895,898,901,903],[955,955,956,956,958,958,959,959,959,959,959,958,958,958,957,957,956],[957,958,960,962,964,967,977,981,991,994,1004,1010,1015,1020,1024,1028,1030,1036,1037,1037,1039,1040,1041,1041,1041,1040,1039,1037,1035,1034,1034,1034,1033,1033,1033,1033,1034,1034,1034,1035,1035,1036,1036],[1177,1178,1181,1182,1183,1183,1181,1179,1173,1170,1160,1155,1143,1138,1130,1119,1116,1110,1110,1109,1111,1114,1116,1118,1124,1128,1131,1134,1141,1144,1147,1154,1156,1157,1157,1151,1148,1141,1129,1125,1111,1107,1104,1096]],"y":[[311,312,315,413,433,440,453,469,475,488,492,501,504,507,508,509,509,509,508,508,506,505,505,504,504,504,504,504,504,504],[308,305,303,302,304,309,319,331,347,363,379,394,399,405,422,428,431,436,437],[401,399,398,397,395,394,392,390,388,387,386,383,379,370,364,358,354,342,339,332,323,321,314,311,306,305,304,304,304,306,308,312,326,332,351,365,382,397,409,415,424,435,438,444,446,448],[332,315,309,305,302,299,295,293,292,289,289,292,296,302,309,313,327,332,347,351,356,364,373,380,384,387,391,400,404,413,415,423,428,434,439,444,447,449,451,450],[354,353,352,351,350,348,347,344,344,342,341,341,341,341,342,344],[417,417,417,417,417,416,415,414,412,410,409,408,407,407,407],[325,317,317,317,320,322,335,341,348,364,382,398,413,419,425,433,437,447,449,451,454,456],[319,317,312,307,303,300,299,297,297,299,302,307,308,311,313,321,323,331,335,345,352,359,366,372,375,379,381,384,385,388,390,393,396,399,404,408,410,415,421,425,429,434,437,439],[318,319,323,327,338,352,366,372,389,394,406,410,418,422,425,426,427],[377,377,374,373,372,371,367,365,362,361,355,351,345,340,335,329,327,318,316,314,310,307,305,304,305,314,317,331,343,354,366,372,389,395,409,414,424,428,431,435,437,438,439],[335,332,328,325,318,314,311,307,301,299,297,297,299,301,305,314,317,328,332,344,351,358,361,365,370,373,376,379,385,388,390,399,404,409,412,419,421,424,429,429,431,431,431,430]],"t":[[0,1,36,66,81,84,92,100,108,117,128,133,144,150,161,167,177,183,194,200,211,217,217,228,233,242,250,252,261,265],[2395,2401,2409,2417,2434,2444,2451,2461,2468,2478,2484,2495,2501,2502,2526,2534,2536,2542,2550],[2704,2709,2716,2720,2726,2734,2742,2751,2761,2767,2768,2778,2784,2809,2811,2825,2828,2835,2842,2851,2859,2868,2876,2884,2895,2901,2912,2918,2928,2934,2935,2945,2951,2961,2968,2978,2985,2993,3001,3009,3018,3026,3034,3045,3051,3059],[3637,3655,3661,3668,3671,3678,3685,3685,3693,3701,3709,3718,3726,3734,3743,3751,3762,3768,3779,3785,3785,3796,3801,3812,3818,3818,3829,3834,3845,3851,3862,3868,3879,3884,3895,3901,3912,3918,3929,3935],[4245,4266,4271,4276,4280,4285,4296,4304,4304,4317,4318,4326,4339,4343,4351,4359],[4553,4560,4568,4576,4585,4596,4601,4602,4613,4618,4629,4635,4646,4652,4656],[10673,10692,10704,10712,10720,10729,10737,10745,10748,10754,10764,10771,10782,10787,10790,10798,10804,10815,10820,10821,10832,10837],[11048,11056,11063,11064,11071,11081,11087,11096,11105,11112,11121,11129,11137,11139,11148,11154,11165,11170,11182,11187,11198,11204,11215,11221,11231,11237,11240,11249,11254,11265,11270,11282,11287,11298,11304,11312,11325,11337,11338,11346,11356,11362,11371,11379],[11609,11628,11633,11637,11646,11654,11664,11671,11682,11687,11698,11704,11715,11721,11732,11737,11748],[11868,11873,11881,11888,11892,11899,11904,11913,11921,11929,11937,11946,11954,11964,11971,11982,11988,11999,12004,12005,12016,12021,12032,12038,12063,12071,12079,12088,12096,12104,12115,12121,12132,12138,12149,12154,12165,12171,12171,12183,12188,12188,12199],[12462,12467,12474,12479,12488,12496,12505,12515,12521,12529,12538,12549,12554,12566,12571,12582,12588,12596,12604,12615,12621,12632,12638,12638,12650,12654,12655,12666,12671,12671,12683,12688,12699,12704,12715,12721,12732,12738,12749,12754,12766,12771,12772,12780]],"version":"2.0.0"}. Así se verifica la identidad.

    Verifica la identidad 1+cotϕcosecϕ=sinϕ+cosϕ donde ϕ{"x":[[469,463,462,461,461,461,463,467,472,479,486,494,502,510,517,525,532,539,546,552,555,557,557,556,551,544,535,525,515,503,491,481,470,460,452,445,439,437,437,437,439,445,452,460,468],[515,514,514,514,514,515,516,516,516,516,516,516,516,516,517]],"y":[[300,304,309,315,321,328,335,340,344,348,350,350,350,346,340,334,326,318,310,301,294,287,282,277,274,272,271,271,271,272,277,283,289,298,306,316,327,337,346,355,361,365,368,368,368],[215,221,229,241,257,273,290,309,329,351,374,396,417,436,452]],"t":[[0,35,45,56,62,72,78,89,95,105,112,122,129,139,145,155,162,172,178,189,195,206,212,222,229,239,247,256,262,275,279,289,295,306,312,322,329,339,345,356,362,372,378,389,395],[732,753,762,774,779,789,795,806,812,822,829,840,849,855,862]],"version":"2.0.0"} está definida sobre el dominio apropiado.

    Solución:

    El LHS de la ecuación parece un poco complicado comparado con el RHS de la ecuación. Siempre es más fácil simplificar el lado más complicado de la ecuación. Así que simplifiquemos el LHS para obtener el RHS.

    Paso 1: Escribiendo el denominador por separado, obtenemos

    LHS =1cosecϕ+cotϕcosecϕ

    Paso 2: Utilizando la identidad recíproca

    sinϕ=1cosecϕ

    obtenemos

    LHS =sinϕ+cotϕcosecϕ

    Paso 3 : Utilizando de nuevo la misma identidad, y cotϕ=cosϕsinϕobtenemos

    LHS =sinϕ+cosϕ{"x":[[122,122,123,124,125,126,126,127,127,127,126,125,123,119,116,111,107,105,103,103,103,105,109,112,116,120,123,125,127,128,128,129,128,127,125,122,118,114,110],[153,153,153,153,153,154,155,155,156,156,157,157,156,155,154,153,152],[149,149,148,150,152],[179,178,178,179,179,180,180,180,180,180,180,180,180,180,180,181,183,184,186,188,189,190,192,193,194,195,196,197,198,198,198,199,200,201,203,205],[229,228,227,227,226,226,226,226,226,227,228,230,233,236,239,243,247,250,254,256,258,258,259,258,256,253,249,244,239,235,231,229],[236,236,234,234,234,234,234,234,235,235,235,235,235,235,235,235,236,237,238],[317,317,318,322,329,338,346,353,361,366,371,376,379,382,384,385],[342,342,342,342,342,342,342,341,340,338,337,335,334,334,334,336],[498,496,494,493,492,491,490,489,488,486,486,485,485,485,487,489,492,495,499,503,507,511,515,518,521,522,523,524,524,523,522,521,519,517,516,514,513,512,512,513,515,517,520,523,526,529,531,533,535,535,536,536,536,536,534,532,530,528,527,525,524,524,524,524,526,529,533,538,545,552,558,565,571,576,581,586,589,593,595,596,597,597,595,593,590,586,582,579,576,574,573,573,573,575,578,581,584,588,591,594,595,596,596,595,593,591,588,587],[617,618,618,619,621,622,624,626,629,631,634,637,640,643,646,649,651,653,653,653,653,650,647,642,637,632,626,622,619,617,616,616,617],[632,631,630,630,631,631,632,633,634,636,637,639,640,642,642,643,643,643,642,642,642]],"y":[[246,245,243,241,241,240,238,237,236,235,233,233,232,232,233,235,237,239,241,243,245,247,249,251,254,256,259,262,264,266,268,270,272,273,275,277,279,280,281],[241,240,239,241,242,244,248,250,256,261,266,271,274,277,278,279,280],[219,217,213,214,216],[248,246,248,251,254,256,259,262,265,267,268,269,268,266,263,260,257,254,251,249,247,247,247,248,250,252,254,257,260,264,267,270,273,274,275,276],[251,250,250,249,249,251,253,257,260,265,269,272,274,276,276,276,274,272,270,267,264,261,258,256,253,252,251,251,252,254,257,258],[229,228,226,227,228,232,237,244,252,261,270,279,286,293,299,304,309,312,315],[266,265,265,265,265,264,263,262,261,260,260,259,258,258,258,258],[250,249,250,251,255,261,268,275,281,287,292,296,299,302,303,304],[242,240,240,240,241,242,245,247,249,254,257,262,267,270,273,275,275,275,275,274,272,269,266,264,260,258,256,254,252,252,251,251,252,254,256,259,262,265,268,271,273,275,277,277,277,277,275,273,271,268,264,260,257,253,251,250,249,249,250,252,253,256,259,261,264,266,268,269,269,269,268,266,264,262,260,257,255,251,249,246,243,241,240,239,239,239,240,241,242,244,245,247,249,252,254,256,258,260,262,264,265,267,268,269,270,270,270,270],[244,245,246,248,251,255,259,263,266,268,269,269,269,267,265,262,259,256,253,250,247,245,244,243,243,243,244,245,246,247,247,248,248],[228,225,224,223,225,228,232,236,241,247,254,261,269,275,282,287,292,297,301,304,306]],"t":[[0,10,19,23,31,32,42,49,60,66,76,83,93,99,110,116,126,133,143,149,160,166,176,183,193,199,209,216,226,233,243,249,260,266,276,283,293,299,309],[494,499,507,519,528,528,536,543,549,559,566,576,583,592,599,609,616],[759,762,768,783,793],[1086,1096,1125,1133,1142,1150,1159,1166,1176,1183,1193,1200,1225,1233,1243,1250,1262,1266,1276,1283,1295,1300,1311,1316,1328,1333,1343,1350,1362,1369,1376,1383,1393,1400,1409,1416],[1685,1698,1703,1709,1716,1741,1750,1760,1766,1776,1783,1793,1800,1810,1817,1826,1833,1843,1850,1860,1866,1876,1883,1893,1900,1913,1916,1927,1933,1943,1950,1959],[2209,2222,2227,2230,2234,2243,2250,2260,2267,2277,2283,2295,2300,2311,2317,2327,2333,2343,2350],[2831,2847,2851,2859,2867,2877,2884,2894,2900,2911,2917,2927,2934,2944,2950,2958],[3166,3186,3190,3194,3201,3212,3217,3227,3234,3244,3250,3260,3267,3277,3284,3294],[3915,3930,3934,3938,3946,3947,3954,3963,3963,3971,3978,3984,3995,4001,4012,4017,4028,4034,4045,4051,4061,4067,4078,4084,4094,4101,4111,4120,4128,4134,4144,4151,4161,4167,4178,4184,4194,4201,4211,4217,4228,4234,4244,4251,4261,4268,4278,4284,4295,4301,4311,4317,4328,4334,4345,4351,4361,4368,4378,4384,4393,4401,4411,4421,4426,4434,4444,4451,4461,4467,4478,4484,4495,4501,4511,4518,4528,4534,4545,4551,4561,4568,4578,4584,4595,4601,4611,4618,4628,4634,4645,4651,4661,4668,4676,4687,4695,4701,4711,4721,4728,4734,4745,4751,4761,4768,4778,4784],[5085,5118,5128,5134,5146,5151,5162,5168,5179,5184,5196,5201,5212,5218,5229,5234,5245,5251,5262,5268,5279,5284,5296,5301,5312,5324,5329,5337,5346,5351,5362,5368,5378],[5536,5543,5549,5562,5585,5595,5601,5613,5625,5629,5635,5646,5652,5662,5668,5679,5685,5695,5701,5712,5718]],"version":"2.0.0"}

    Así podemos ver que LHS=RHS{"x":[[268,269,270,270,271,271,271,270,268,264,263,260,259,258,256,256,255,254,254,254,256,257,262,267,273,276,279,283,296,300,313,317,330,337,346,348,352,355,362],[379,379,379,380,380,380,380,380,379,377,376,374,373,372,372,373,374,375],[388,388,388,388,389,390,393,399,405,411,414,423,428,433,438,440,445,447,451,452,454,454,454,454,454,453,451,450,449,447,446,444,443,443,443,442,442,441,441,441,442,443,444,449],[583,586,588,590,590,590,590,586,582,577,573,570,562,557,554,546,538,535,529,528,528,528,534,536,545,551,558,564,567,574,576,576,577,577,576,571,565,562,555,550,546,532,528,516,513,510],[668,670,673,674,683,686,697,701,712,715,726,731,733,738,740],[677,677,676,680,682,690,694,698,707,714,721,726],[843,844,846,847,847,847,847,843,842,840,836,834,832,830,825,823,821,817,816,816,815,815],[840,841,844,846,855,858,867,870,879,884,887,890,891,891,890,886,881,876,873,863,857,855,848,847,847,848,851,854,857,860,866,870,873,886,892,898,902,907,908,911,915,916,918,919,920,920],[977,978,980,981,981,981,982,982,980,978,977,976,973,971,969,966,966,964,964,964,964,964],[963,962,963,967,972,976,988,994,1006,1010,1021,1027,1032,1036,1038,1042,1043,1046,1046,1046,1046,1044,1043,1042,1041,1036,1034,1029,1028,1023,1020,1018,1017,1017,1016,1016,1016,1016],[1089,1091,1093,1097,1100,1102,1106,1106,1106,1103,1100,1094,1088,1083,1080,1069,1066,1057,1055,1051,1051,1051,1051,1054,1056,1063,1064,1070,1073,1075,1076,1075,1073,1070,1062,1058,1046,1036,1027,1018,1009,1006]],"y":[[248,248,249,250,255,263,269,290,297,318,324,344,351,358,372,378,383,389,404,412,418,420,425,427,427,427,427,425,421,419,414,412,408,405,403,402,402,401,399],[267,266,265,269,272,277,287,301,309,328,334,349,353,364,367,371,371,371],[329,328,327,326,325,324,322,319,317,314,313,308,304,299,294,292,284,281,271,268,260,258,256,252,251,247,246,247,248,256,264,273,284,288,306,312,331,342,352,360,367,370,374,378],[279,269,264,258,255,248,245,240,238,238,238,239,243,245,248,255,263,268,279,283,294,297,305,309,317,322,327,332,335,342,344,347,351,354,356,360,364,365,368,369,369,370,370,368,367,365],[292,291,289,289,288,288,288,288,288,288,289,289,290,291,292],[338,339,339,340,340,338,338,337,335,333,332,331],[249,248,244,242,241,245,247,261,266,272,287,295,302,311,327,335,341,357,365,370,374,375],[257,254,248,245,239,238,235,234,234,236,237,242,244,251,254,262,269,275,278,288,294,296,303,305,308,309,311,312,312,314,315,316,317,321,325,329,332,338,340,343,351,353,358,364,366,367],[241,239,237,233,232,231,231,237,244,253,258,264,277,284,298,314,318,328,331,337,338,339],[294,294,294,292,290,289,285,284,279,277,271,267,261,255,253,244,241,234,232,227,226,225,225,225,226,233,237,249,253,268,278,289,298,302,312,315,318,321],[252,251,249,245,242,241,236,235,231,230,229,229,231,234,235,242,244,253,256,265,267,269,272,278,280,286,288,293,297,301,304,308,311,312,317,318,320,321,321,321,320,319]],"t":[[0,41,47,58,64,73,81,89,97,109,114,123,131,135,142,147,148,159,164,175,181,189,198,208,214,223,226,231,242,247,259,264,273,281,289,298,302,309,314],[523,528,535,556,564,568,575,581,592,598,608,615,623,634,639,648,656,667],[762,773,781,792,798,806,815,825,831,842,848,858,864,875,881,892,898,909,914,925,931,939,942,948,958,965,981,992,998,1008,1015,1025,1031,1042,1048,1058,1064,1073,1081,1092,1098,1109,1115,1125],[1375,1392,1399,1406,1416,1423,1431,1440,1448,1459,1465,1465,1476,1481,1481,1492,1498,1509,1515,1525,1531,1542,1548,1558,1565,1575,1581,1592,1598,1609,1615,1615,1626,1633,1634,1647,1649,1656,1665,1670,1673,1681,1692,1698,1709,1713],[2059,2080,2084,2090,2098,2109,2115,2126,2131,2142,2148,2159,2165,2173,2182],[2300,2307,2316,2340,2348,2359,2365,2365,2376,2382,2392,2398],[2621,2629,2638,2638,2646,2657,2665,2673,2682,2684,2693,2698,2699,2707,2715,2717,2726,2732,2743,2748,2759,2765],[2912,2917,2924,2932,2943,2949,2959,2965,2976,2982,2993,2998,3010,3015,3024,3032,3042,3049,3060,3065,3076,3082,3093,3099,3110,3115,3124,3132,3135,3140,3149,3151,3160,3165,3176,3182,3193,3199,3199,3210,3223,3224,3232,3243,3248,3257],[3417,3424,3428,3433,3441,3449,3457,3465,3476,3482,3490,3491,3499,3509,3515,3526,3532,3543,3549,3560,3565,3566],[3680,3699,3715,3726,3732,3743,3749,3760,3765,3777,3782,3793,3799,3810,3816,3827,3832,3843,3849,3857,3866,3876,3882,3882,3894,3899,3907,3916,3926,3932,3943,3949,3957,3966,3976,3982,3985,3994],[4159,4166,4170,4176,4183,4191,4199,4207,4216,4224,4232,4243,4249,4260,4266,4277,4282,4294,4299,4311,4316,4316,4327,4333,4344,4349,4360,4366,4374,4382,4393,4399,4411,4416,4427,4433,4444,4449,4461,4466,4477,4482]],"version":"2.0.0"}.

    Así queda demostrada la identidad.

    ¿Cuándo es inválida una identidad trigonométrica?

    Una identidad trigonométrica es verdadera si y sólo si satisface todos los valores para los que está definida la función. Para demostrarlo, considera la identidad

    sin2x=cosx{"x":[[264,289,290,290,289,288,287,284,282,278,275,263,259,255,253,250,249,249,248,248,249,250,252,254,256,259,261,264,267,270,273,274,276,277,277,277,275,273,270,266,263,259,254,250,246,241,238,234,230,227,225,223,222,223],[303,303,303,303,303,303,303,303,303,302,302,301,301,300,299,298,298],[311,311,311,311,312],[329,329,329,329,329,329,329,329,329,329,329,328,328,327,327,326,326,326,327,328,329,330,333,335,337,339,340,341,342,342,342,343,343,343,343,343,344,344,344,344,345,347,349,351,354],[353,352,351,351,351,351,352,353,355,356,358,360,362,363,365,366,366,367,367,366,365,365,364,363,362,362,362,363,364,367,369,373,377,379],[389,389,388,387,387,386,386,386,386,387,389,391,393,395,397,398,399,400,400,399,398,397,396,395,394,395,396,399,401,404,406,409,412,414,416,417,418,417,416,414,413,412,411,410,410,411,413,417,421,425,428],[491,492,493,494,497,501,505,509,513,516,520,523,525,528,530,531,532,532],[498,497,496,496,497,500,504,509,515,521,527,532,536,540,542,544],[610,610,610,610,610,609,607,606,604,602,599,597,594,592,590,589,589,590,592,595,597,600,604,608,612,617,620,623,626,628,630,631,631,631,631,630,630,630,630,630,630,630,631,632,633,635,636,638,639,641,642,643,645,645,646,646,646,646,644,642,640,637,635,633,631,629,627,626,626,626,626,628,630,633,636,640,644,649,654,659,664,668,673,677,681,684,686,688,689,689,689,688,686,684,681,679,677,675,674,674,673,673,675,676,678,681,684,687,691,694,696,698,699,699,699,699,697,695,692,689,685,682,679],[713,713,713,714,714,716,718,721,724,726,728,731,733,735,737,738,738,738,737,735,732,730,728,727,726,725,726,727,728,730,733,735,738,741,743,745,747,749,750,750,750,749,749,748,748,748,748,749,752,755,761,767,774,781]],"y":[[266,236,233,230,229,228,227,227,227,228,229,236,239,242,245,249,251,253,255,257,259,260,261,262,263,263,264,265,266,267,268,270,271,273,275,277,280,283,286,290,293,296,299,301,302,303,303,303,303,302,301,299,297,294],[253,252,251,252,254,257,262,266,272,278,282,286,289,291,293,293,294],[233,232,231,230,230],[261,260,259,258,257,258,259,262,265,268,272,275,279,282,284,286,287,286,285,282,279,274,270,265,262,259,259,260,262,264,267,270,274,277,280,283,287,289,291,293,294,294,294,294,293],[226,224,220,219,218,216,214,213,211,209,207,206,205,205,205,206,208,210,213,215,218,221,223,226,227,228,229,229,229,229,228,226,225,224],[268,267,266,266,265,264,263,262,261,260,259,258,258,258,259,261,263,266,269,272,275,278,280,282,283,283,281,279,276,273,270,267,264,262,260,259,259,261,263,266,270,273,276,280,283,285,287,289,289,289,288],[249,249,249,248,247,247,246,245,244,244,244,244,244,244,245,245,246,248],[263,263,263,262,262,262,262,262,261,261,260,259,258,257,257,257],[247,246,245,244,243,242,242,242,242,242,244,247,251,255,260,264,268,271,273,275,276,276,276,275,274,272,271,269,266,264,263,261,259,258,257,257,258,260,262,265,267,270,273,276,278,279,280,280,280,279,278,276,273,270,267,264,261,258,255,253,252,252,252,253,254,256,259,262,264,267,270,272,274,275,276,276,276,275,274,272,270,268,265,263,261,258,256,253,251,249,247,245,243,243,242,242,242,243,243,244,245,247,249,251,253,255,257,259,261,263,265,267,269,270,271,272,273,273,273,273,273,272,270],[251,250,249,248,247,247,246,245,245,245,245,245,246,247,249,251,254,257,261,264,267,270,272,273,274,274,274,273,271,269,267,264,261,257,254,251,248,247,246,248,250,253,256,259,262,265,268,270,273,274,274,274,272,270]],"t":[[0,6,12,23,31,41,47,57,64,74,81,107,118,125,131,142,147,158,164,175,181,192,197,208,214,225,231,241,247,258,264,275,281,292,298,309,314,325,331,342,347,358,364,375,381,392,397,408,414,425,431,442,448,458],[827,841,856,866,876,881,892,898,909,914,925,931,942,948,959,964,976],[1133,1138,1144,1146,1172],[1511,1521,1527,1529,1540,1573,1581,1593,1598,1610,1615,1626,1631,1643,1648,1659,1665,1690,1698,1709,1715,1726,1731,1742,1748,1759,1765,1776,1781,1792,1798,1809,1815,1826,1831,1842,1848,1859,1865,1876,1881,1893,1898,1909,1915],[2193,2203,2209,2211,2217,2226,2232,2243,2248,2260,2265,2276,2282,2293,2298,2309,2315,2326,2332,2343,2348,2359,2365,2376,2382,2393,2398,2410,2415,2426,2432,2443,2448,2456],[2852,2862,2867,2869,2876,2890,2898,2910,2927,2932,2944,2948,2960,2965,2976,2982,2993,2998,3010,3015,3027,3032,3043,3048,3060,3090,3099,3110,3115,3127,3132,3143,3148,3160,3165,3177,3182,3215,3225,3232,3243,3249,3260,3265,3277,3282,3293,3299,3310,3315,3323],[3895,3900,3917,3926,3932,3944,3949,3961,3965,3977,3982,3994,3999,4010,4015,4027,4032,4044],[4209,4214,4222,4228,4241,4249,4261,4266,4278,4282,4294,4299,4311,4316,4327,4332],[5033,5039,5044,5047,5058,5066,5083,5094,5099,5111,5116,5128,5133,5145,5149,5161,5166,5178,5183,5195,5199,5211,5216,5227,5233,5244,5249,5261,5266,5277,5283,5294,5299,5311,5333,5344,5361,5366,5378,5383,5395,5399,5411,5416,5428,5433,5444,5449,5461,5466,5478,5483,5494,5499,5511,5516,5528,5533,5545,5549,5561,5566,5578,5583,5595,5599,5611,5616,5624,5633,5644,5650,5661,5666,5678,5683,5694,5699,5711,5716,5728,5733,5745,5749,5761,5766,5778,5783,5795,5799,5811,5816,5828,5833,5844,5849,5861,5866,5878,5883,5895,5899,5911,5916,5928,5933,5945,5950,5961,5966,5978,5983,5995,5999,6011,6016,6028,6033,6045,6050,6061,6066,6082],[6436,6442,6448,6450,6458,6464,6470,6479,6486,6495,6496,6503,6512,6516,6528,6533,6545,6550,6562,6566,6578,6583,6595,6600,6612,6616,6642,6650,6661,6667,6679,6683,6695,6700,6712,6717,6729,6733,6745,6775,6783,6795,6800,6812,6817,6829,6833,6845,6850,6862,6867,6875,6883,6895]],"version":"2.0.0"}

    Esta identidad no se cumple para todos los valores de x en el dominio del seno y el coseno. Sólo es cierta para determinados valores de x. Para demostrar que esta identidad es falsa, se puede demostrar algebraicamente que no es válida o dar un contraejemplo. Un contraejemplo es

    LHS=sin245°=12

    RHS=cos45°=12

    Donde puede verse que LHSRHS{"x":[[213,213,200,190,180,160,185,195,223,253,270,284,296,300,307,310,314,316],[318,319,320,320,321,323,323,323,323,321,321,318,317,316,316,314,313,312,311,311,310],[330,331,331,333,334,341,344,354,357,363,373,376,385,388,396,398,406,412,415,416,418,420,420,421,421,421,420,419,418,417,416,413,413,412,411,411,412,413,413,415,417,419,420,422,423,426,426],[564,565,566,566,566,564,563,558,556,550,547,531,517,511,508,502,501,500,504,506,513,517,524,534,538,547,550,557,561,561,556,554,544,541,530,514,510,501,500,496],[657,658,660,664,670,674,683,696,701,717,721,727,737,754,763,766,773,776,780,780,781],[687,686,687,688,691,700,704,719,724,735,751,769,774,781,784,788,791,792],[737,737,737,737,737,737,737,736,736,733,731,728,727,725,721,720,718,717,715,714,713,712,712,712,713,714,716],[903,903,904,904,905,905,906,905,902,899,896,894,890,889,886,884,883,883,882,882,883,884,886,887],[905,906,907,909,911,919,923,935,941,944,949,951,953,956,956,954,953,948,945,938,934,923,919,907,904,896,894,892,892,892,893,898,900,903,910,913,923,927,936,938,943,948,950,956,959,960,961,962,963,967,968,970],[1047,1048,1048,1048,1048,1047,1047,1046,1045,1044,1044,1043,1043,1043,1043,1044,1045],[1044,1044,1043,1041,1041,1041,1042,1043,1047,1048,1055,1060,1063,1068,1074,1078,1084,1087,1092,1098,1103,1107,1110,1111,1116,1117,1120,1122,1123,1124,1123,1123,1123,1122,1121,1120,1119,1117,1117,1116,1116,1116,1116,1116,1116,1116,1116,1117,1117,1118,1118,1119,1120,1121],[1263,1266,1267,1267,1265,1264,1257,1253,1240,1236,1225,1209,1204,1191,1189,1186,1186,1190,1192,1198,1209,1213,1225,1228,1237,1239,1243,1243,1243,1241,1238,1233,1219,1198,1186,1174,1168,1158,1150]],"y":[[208,209,299,332,365,460,472,472,468,466,465,464,464,464,464,464,462,460],[220,220,221,224,227,243,250,272,291,309,315,332,345,354,359,372,376,382,390,391,393],[322,321,320,318,317,314,313,311,311,310,309,308,305,304,299,297,291,281,271,268,258,248,233,215,209,204,204,204,204,209,213,226,231,241,258,286,294,314,320,338,352,364,368,377,378,382,383],[235,233,228,226,221,219,218,217,216,217,218,225,237,243,247,259,263,274,287,291,299,303,309,318,321,330,333,342,354,365,374,377,385,388,393,396,396,393,392,388],[317,317,317,316,316,316,316,317,317,317,317,317,318,318,318,319,319,320,322,323,325],[365,367,367,367,367,367,367,367,367,366,365,364,364,364,364,365,365,366],[281,279,277,275,272,271,272,279,283,298,309,324,331,345,367,372,390,395,411,415,428,433,435,439,440,440,438],[271,269,267,266,267,268,275,289,310,326,342,351,373,381,401,407,417,425,429,432,442,444,444,442],[271,268,265,258,256,248,246,242,241,241,243,245,247,256,259,270,274,282,287,295,299,311,315,325,328,335,337,340,341,342,345,349,351,352,356,358,365,368,378,381,389,399,402,411,418,419,421,422,423,424,424,422],[271,273,276,291,297,318,325,331,342,348,357,371,375,382,385,388,395],[357,356,354,351,347,346,342,341,338,338,335,334,332,331,328,327,324,322,319,315,311,306,303,301,292,289,281,277,272,268,266,265,266,267,271,277,280,289,298,308,315,327,339,346,353,369,375,389,393,399,402,405,409,412],[277,270,262,258,257,256,255,255,257,259,264,275,279,292,297,309,312,323,326,332,341,344,354,357,367,371,382,386,390,398,401,408,418,427,431,432,433,433,431]],"t":[[0,1,17,34,51,122,145,159,174,198,205,221,230,238,246,255,268,282],[740,755,764,776,780,791,796,806,822,830,838,847,855,866,875,887,889,897,911,913,922],[1106,1114,1122,1130,1138,1147,1155,1163,1172,1180,1188,1197,1205,1213,1222,1231,1238,1255,1272,1286,1296,1307,1314,1340,1347,1358,1363,1372,1383,1393,1397,1408,1414,1422,1430,1447,1455,1467,1473,1485,1497,1507,1514,1522,1530,1538,1547],[1912,1912,1915,1922,1935,1939,1950,1955,1967,1972,1972,1992,2009,2014,2023,2030,2039,2047,2064,2072,2083,2092,2104,2106,2114,2122,2134,2139,2155,2172,2183,2193,2204,2207,2214,2230,2239,2250,2255,2267],[2574,2581,2589,2602,2605,2614,2622,2631,2639,2647,2656,2656,2664,2680,2689,2700,2707,2714,2722,2731,2739],[2896,2907,2931,2932,2939,2949,2956,2964,2972,2981,2989,3011,3020,3022,3023,3033,3039,3052],[3266,3271,3278,3281,3282,3289,3314,3322,3331,3341,3347,3356,3364,3372,3384,3389,3401,3411,3420,3423,3431,3439,3451,3456,3467,3472,3486],[3762,3767,3775,3782,3789,3801,3806,3823,3831,3839,3848,3856,3864,3873,3884,3889,3901,3906,3906,3917,3931,3939,3948,3956],[4132,4139,4145,4148,4156,4168,4173,4184,4189,4201,4206,4206,4217,4224,4231,4239,4248,4256,4258,4268,4273,4284,4291,4301,4306,4320,4327,4331,4331,4342,4348,4356,4364,4365,4373,4381,4389,4398,4406,4417,4425,4431,4439,4448,4465,4465,4473,4476,4485,4490,4502,4506],[4688,4714,4723,4735,4740,4751,4756,4760,4768,4773,4785,4790,4802,4806,4806,4815,4823],[4933,4941,4946,4951,4956,4968,4973,4985,4990,5001,5006,5018,5023,5035,5040,5040,5052,5057,5068,5073,5085,5090,5090,5102,5107,5118,5123,5135,5140,5156,5168,5173,5185,5190,5202,5207,5207,5219,5224,5232,5240,5251,5256,5257,5269,5274,5285,5291,5301,5307,5307,5319,5324,5331],[5603,5619,5627,5628,5634,5640,5652,5657,5669,5673,5685,5690,5702,5707,5719,5723,5735,5740,5752,5757,5769,5774,5785,5790,5802,5807,5815,5823,5827,5835,5840,5852,5857,5873,5885,5890,5902,5907,5927]],"version":"2.0.0"}.

    Hasta ahora hemos visto cómo se puede demostrar algebraicamente una identidad trigonométrica utilizando las identidades fundamentales. Es importante tener en cuenta que algunas identidades pueden parecer verdaderas, pero en realidad son falsas. Esto tiene que ver con que una función esté indefinida para un determinado valor del dominio, mientras que la otra función esté definida. En ese caso, cuando dividimos por algo, el denominador puede tender a 0 y esa identidad sería falsa. Es una situación poco frecuente, pero hay que tenerla en cuenta.

    Resolver una identidad algebraicamente no es la única forma de demostrarla, una forma alternativa es demostrarla gráficamente, veámoslo.

    ¿Cómo demostrar gráficamente una identidad trigonométrica?

    Para demostrar gráficamente una identidad, verificamos el LHS y el RHS por separado. En primer lugar, se representa la gráfica de la LHS y, a continuación, la de la RHS. Tras examinar las gráficas, si son idénticas, es decir, exactamente iguales para cada valor de su dominio, diremos que LHS=RHS{"x":[[209,209,209,209,209,208,207,207,205,204,202,201,200,198,197,196,194,191,190,188,188,189,191,196,201,207,211,216,220,236,248,259,271,284,294,305,314,318,321],[336,336,335,335,336,336,337,337,337,337,337,336,336,336,336,337,338,340],[342,342,344,348,351,363,367,378,386,393,396,399,406,410,412,418,420,424,427,429,430,431,431,431,431,431,430,429,428,426,425,424,423,421,421,421,421,421,421,421,421,421,423,424,426],[576,566,564,556,548,540,532,527,517,513,508,506,506,505,508,512,518,525,532,536,547,550,559,561,563,565,566,566,566,562,560,549,544,529,520,511,508],[666,668,670,681,691,701,707,711,715,727,731,740,743,744],[668,669,670,671,680,685,702,708,723,728,741,745,748,756,762,764],[848,852,853,854,854,853,853,852,850,848,847,845,845,845,845,846,847,850,850],[854,855,857,860,864,871,875,887,890,901,905,912,914,916,918,919,919,918,916,913,904,900,888,881,874,870,863,859,858,856,855,855,855,857,860,861,864,866,874,877,888,893,899,902,907,912,913,916,916,917],[969,970,970,970,972,973,973,973,971,971,969,968,967,967,966,966,966,966,965],[963,964,967,971,974,977,980,988,991,1000,1002,1009,1013,1017,1018,1020,1023,1026,1028,1032,1033,1036,1037,1038,1038,1038,1038,1037,1036,1035,1033,1032,1031,1029,1028,1027,1026,1023,1023,1022,1021,1021,1021,1021,1021,1021,1021,1022,1022,1023],[1193,1198,1198,1200,1200,1197,1196,1194,1191,1184,1176,1167,1163,1151,1143,1139,1130,1127,1124,1124,1124,1127,1129,1131,1134,1143,1146,1156,1160,1167,1171,1174,1174,1173,1171,1168,1166,1153,1147,1130,1123,1116,1104,1098]],"y":[[227,229,235,240,246,261,279,287,312,321,347,356,365,385,393,402,411,437,444,465,471,485,492,497,501,502,503,503,503,502,500,498,497,495,494,492,491,491,491],[296,297,306,312,337,345,365,372,393,407,414,432,438,449,453,458,460,460],[394,393,391,388,387,379,377,371,367,363,361,359,355,350,348,340,337,327,319,311,303,299,285,282,278,277,274,274,277,282,292,304,311,334,349,366,382,395,402,421,426,435,447,449,451],[289,281,281,282,285,290,296,299,311,315,325,330,334,347,354,361,368,374,380,383,392,394,402,405,408,413,415,417,419,426,428,434,436,440,441,441,439],[341,341,341,341,341,340,340,339,339,339,339,339,339,339],[404,404,404,404,404,404,401,400,397,397,395,394,394,393,392,391],[252,268,290,297,320,337,352,361,388,395,412,434,439,454,457,464,465,465,463],[253,250,244,237,232,227,225,222,221,222,224,231,235,239,247,252,257,267,278,282,299,305,319,329,338,341,351,357,358,364,366,368,372,376,381,382,385,387,394,396,404,408,414,417,424,435,439,449,451,454],[269,270,271,272,284,291,315,322,343,349,367,378,387,392,395,399,407,409,412],[371,371,370,369,369,368,368,366,365,362,361,355,351,346,344,340,334,326,322,311,307,297,290,285,279,277,272,271,269,269,272,275,279,288,293,299,304,321,328,349,362,376,381,392,408,412,425,429,438,441],[299,288,285,281,276,272,270,269,268,267,267,269,271,278,284,287,299,304,315,319,323,331,334,338,341,352,355,366,369,379,387,394,400,406,409,412,414,420,422,425,425,425,425,424]],"t":[[0,23,34,39,40,50,56,67,73,84,89,98,98,106,114,115,123,134,139,150,156,167,173,184,189,200,206,206,214,223,234,239,250,256,267,273,284,289,294],[549,573,581,589,598,606,615,623,633,639,650,656,667,673,681,690,700,706],[837,865,873,881,890,898,906,914,923,934,940,940,950,956,967,973,984,990,998,1006,1017,1023,1040,1042,1048,1056,1065,1073,1083,1090,1098,1106,1117,1123,1134,1140,1150,1157,1167,1173,1184,1190,1200,1206,1211],[1479,1502,1506,1515,1523,1532,1540,1548,1558,1565,1573,1578,1581,1590,1600,1607,1617,1623,1634,1640,1650,1657,1667,1673,1674,1684,1690,1691,1701,1707,1717,1723,1734,1740,1750,1757,1767],[2075,2100,2107,2117,2123,2134,2140,2140,2151,2157,2167,2173,2182,2186],[2338,2345,2352,2357,2365,2374,2384,2390,2401,2407,2418,2423,2424,2434,2440,2448],[2726,2745,2757,2767,2774,2784,2790,2801,2807,2818,2824,2834,2841,2851,2857,2868,2874,2884,2890],[3079,3084,3092,3099,3107,3118,3124,3134,3140,3152,3157,3168,3174,3174,3182,3191,3193,3201,3207,3218,3224,3235,3240,3251,3257,3268,3274,3284,3291,3301,3307,3308,3318,3324,3335,3341,3341,3352,3357,3366,3374,3384,3390,3402,3407,3417,3424,3435,3441,3445],[3680,3685,3699,3710,3716,3724,3732,3741,3751,3757,3768,3774,3785,3790,3791,3802,3807,3816,3824],[3959,3991,3999,4008,4016,4016,4024,4035,4041,4052,4057,4066,4074,4085,4091,4091,4102,4107,4119,4124,4135,4141,4151,4157,4169,4174,4185,4191,4202,4210,4218,4224,4225,4236,4241,4241,4249,4257,4268,4274,4285,4291,4301,4308,4319,4324,4333,4341,4352,4357],[4659,4672,4679,4683,4691,4702,4707,4708,4719,4724,4735,4741,4752,4758,4769,4774,4785,4791,4802,4808,4808,4819,4824,4825,4836,4841,4852,4858,4866,4874,4885,4891,4903,4908,4916,4920,4925,4936,4941,4952,4958,4958,4970,4975]],"version":"2.0.0"}. El principal inconveniente de utilizar este método es que sólo se pueden demostrar las identidades sencillas. Las identidades con potencias múltiples y superiores pueden ser muy difíciles de graficar. Ésta es la razón por la que se prefiere el método algebraico a éste. Vamos a demostrar una identidad para hacernos una idea de este método.

    Prueba que sinx=1cosecx{"x":[[128,128,129,131,133,135,139,142,146,150,154,157,161,163,165,167,169,169,170,170,169,167,165,162,159,156,153,150,148,146,145,144,144,144,145,147,149,151,154,159,160,162,163,163,163,162,160,157,153,149,144,138,133,127,122,117,112,109,108],[188,188,188,188,188,188,188,188,188,187,187,186],[198,197,196,195,195,195,196,196],[207,207,207,207,207,207,207,207,208,208,208,209,209,209,209,209,208,208,207,207,206,206,206,207,209,210,212,215,217,220,222,223,225,226,227,228,229,230,231,233,234,235,237],[255,254,254,254,254,254,254,255,257,260,262,266,269,271,273,274,274,274,274,272,270,268,267,266,265,265,266,267,269,270,272,274,275,277,277,278,279,279,279,279,279,279,278,278,279,280,282,286,290,295,300],[351,350,348,350,353,357,361,366,371,375,379,382,383],[342,344,345,351,359,368,377,385,392,397,402,405,407],[568,569,570,571,572,573,575,576,577,578,578,578,578,578,578,577,576,576,576,576,576],[481,481,482,484,489,496,506,517,528,539,552,565,578,592,606,621,636,650,664,677,688,699,708,716,723,728,731,734,735,735,735,734],[488,488,488,489,489,489,489,488,487,485,483,480,478,475,473,472,471,471,472,473,476,478,481,485,488,491,494,498,501,503,506,508,509,509,510,509,508,508,507,507,507,506,506,506,507,507,509,510,511,513,515,517,518,520,521,521,522,522,521,519,518,515,513,511,509,507,506,506,506,506,508,510,513,517,521,526,532,538,543,548,553,557,561,564,566,568,570,570,570,570,569,568,567,565,564,563,563,562,562,562,562,563,564,566,567,569,569,569,569,568,566,564,561,559,557,556,555],[586,586,585,586,587,588,589,591,592,594,596,597,599,599,600,600,599,597,596,594,592,591,589,588,587,587,587,589,591,594,597,600,604],[638,638,638,638,638,637,636,634,631,628,626,623,621,619,618,618,618,620,622,626,630,635,637],[669,670,670,670,671,671,673,674,677,679,681,683,684,686,687,689,690,692,693,694,694,694,694,693,691,690,688,687,686,685,684,684,685,685,686,687,688,689,691,693,695,698,701,704,707,710,711,713,713,714,714,713,712,711,710,708,707,705,704,703,702,702,704,707,711,716,721,727]],"y":[[223,224,224,223,221,219,217,214,211,208,204,200,196,192,188,184,181,178,176,175,174,174,175,176,178,181,184,188,191,195,200,204,208,212,215,218,220,222,225,229,232,234,237,240,243,245,248,251,253,255,257,258,259,260,260,258,256,253,252],[211,212,214,219,224,230,235,240,243,245,247,247],[182,181,180,179,178,179,180,181],[219,218,217,216,215,214,213,212,212,213,215,219,222,226,230,233,236,237,238,239,238,236,233,230,226,223,220,217,215,214,214,215,217,220,224,227,231,234,237,240,242,243,243],[222,222,221,220,219,218,217,215,213,211,210,209,209,210,212,215,218,222,225,228,232,234,236,237,237,235,233,230,228,225,222,219,216,213,212,211,211,212,213,216,219,222,226,229,232,234,235,236,235,234,231],[206,206,206,206,206,205,204,203,202,201,201,200,200],[223,223,223,222,221,219,217,214,213,211,210,209,209],[133,132,131,129,129,129,129,132,136,141,146,152,159,164,169,174,178,180,183,184,185],[234,233,233,233,233,233,232,231,230,229,228,227,227,227,226,226,225,225,224,224,224,224,224,225,226,226,228,229,230,231,232,233],[295,294,292,291,290,289,287,287,286,286,287,289,292,296,301,306,311,315,318,321,323,324,324,324,324,322,320,318,315,313,310,308,306,304,303,303,304,305,308,310,313,316,319,322,324,326,327,327,327,326,324,321,318,315,312,309,305,302,300,298,297,296,296,297,298,300,303,305,308,310,313,315,316,317,318,317,316,315,313,311,309,307,304,302,299,297,294,292,290,288,287,286,285,285,285,285,286,288,290,292,295,298,301,304,307,309,311,313,314,314,314,314,313,312,311,309,309],[299,298,298,298,299,299,299,299,299,298,296,295,293,291,289,288,287,287,286,287,288,291,293,297,300,304,306,309,310,311,311,311,310],[291,290,289,288,287,287,286,286,286,287,289,291,294,297,301,304,307,309,311,312,312,312,311],[303,303,302,301,300,298,297,295,293,291,290,290,290,291,292,293,295,297,299,302,304,307,310,313,315,318,320,322,323,325,325,326,326,325,325,324,322,321,319,316,314,311,308,306,303,301,299,297,296,295,294,293,293,294,295,298,301,304,307,311,315,318,322,324,325,325,325,323]],"t":[[0,8,39,41,53,57,67,74,84,91,101,107,119,124,140,141,152,158,168,174,184,191,201,207,218,224,234,241,252,257,268,274,284,291,301,308,319,324,336,349,358,368,374,384,391,401,408,416,424,433,442,452,457,468,474,485,491,501,507],[911,933,943,954,958,968,974,984,991,1001,1008,1015],[1179,1189,1193,1200,1208,1218,1224,1232],[1533,1538,1542,1544,1553,1558,1569,1575,1584,1591,1601,1608,1618,1625,1636,1641,1656,1658,1669,1675,1691,1701,1708,1718,1725,1736,1741,1755,1758,1769,1775,1785,1791,1801,1808,1818,1825,1836,1841,1852,1858,1869,1875],[2155,2175,2188,2203,2208,2218,2225,2236,2242,2250,2259,2269,2275,2285,2292,2302,2308,2318,2325,2337,2341,2352,2358,2372,2375,2408,2417,2425,2437,2442,2452,2458,2471,2475,2486,2492,2508,2525,2535,2542,2552,2559,2572,2575,2586,2592,2602,2609,2618,2625,2638],[3134,3140,3146,3167,3175,3185,3192,3202,3209,3219,3225,3235,3242],[3401,3412,3418,3425,3436,3442,3453,3459,3469,3475,3485,3492,3502],[3966,3971,3977,3979,3984,3992,4004,4009,4019,4025,4036,4042,4052,4059,4069,4075,4086,4092,4102,4109,4119],[4588,4601,4609,4619,4626,4636,4643,4651,4659,4669,4676,4686,4692,4703,4709,4719,4726,4736,4743,4753,4759,4769,4776,4786,4792,4803,4809,4822,4826,4837,4843,4850],[5280,5287,5293,5296,5302,5309,5320,5326,5336,5343,5353,5359,5370,5376,5387,5393,5403,5409,5420,5426,5436,5443,5454,5459,5470,5476,5487,5493,5503,5509,5520,5526,5537,5543,5553,5570,5593,5603,5609,5620,5626,5637,5643,5653,5659,5670,5676,5684,5693,5703,5710,5720,5726,5737,5743,5753,5760,5770,5776,5787,5793,5803,5809,5820,5826,5837,5843,5853,5860,5870,5876,5887,5893,5904,5910,5920,5926,5937,5943,5954,5959,5970,5976,5987,5993,6001,6010,6018,6026,6037,6043,6054,6060,6070,6076,6087,6093,6104,6110,6120,6126,6137,6143,6154,6160,6170,6176,6187,6193,6204,6210,6220,6226,6237,6243,6253,6259],[6536,6546,6551,6585,6603,6610,6620,6627,6637,6643,6654,6660,6671,6676,6687,6693,6704,6710,6721,6727,6737,6743,6754,6760,6771,6777,6787,6793,6804,6810,6821,6827,6838],[7212,7221,7227,7235,7243,7254,7260,7271,7277,7288,7293,7304,7310,7321,7327,7338,7343,7354,7360,7371,7377,7388,7393],[15503,15515,15539,15546,15557,15563,15574,15580,15591,15597,15608,15613,15624,15630,15641,15647,15657,15663,15674,15680,15691,15697,15708,15713,15724,15730,15741,15747,15758,15763,15774,15790,15807,15813,15824,15830,15841,15847,15858,15863,15874,15880,15891,15897,15908,15913,15924,15930,15941,15947,15958,15974,15980,15991,15997,16008,16013,16024,16030,16041,16047,16058,16063,16074,16080,16091,16097,16108]],"version":"2.0.0"}.

    Solución:

    Paso 1: Traza la gráfica del LHS, aquí es sinx, y su gráfica tiene el aspecto :

    Verificación de identidades trigonométricas, La gráfica de y=sinx, StudySmarterLa gráfica de y=sinx, StudySmarter Originals

    Paso 2: Traza la gráfica dey=1cosec xfrente a x, y su gráfica se parece a

    Verificar identidades trigonométricas, La gráfica del recíproco de cosecx, StudySmarterLa gráfica derecíproco de cosecx, StudySmarter Originals

    Se puede observar que las gráficas anteriores son exactamente iguales y, por tanto, deducimos que sinx=1cosecx.

    Ejemplos de verificación de identidades trigonométricas

    Demuestra que cos4θ-sin4θ=cos2θ{"x":[[115,117,117,117,117,116,116,114,113,111,109,106,104,101,98,95,93,92,92,93,94,97,99,102,106,109,113,117,121,125,129,132,136,138,141,143,144,145,145,145,144,143,142,141,140,139,138,138,138,138,138,139,140,142,144,145,147,148,149,150,151,153,154,155,156,156,156,155,154,152,151,149,147,145,144,144,144,146,148,151,155,160,166,172,180,186,192,197,201,204,207,208,209,210,210,210,210,209,208,206,204,203,201,200,199,199,199,200,202,205,208,212,214,217,218,218,217,216,213,211,208,205,203,200,199],[213,213,213,213,214,214,214,213,212,211,210,208,207,206,207,208,209,211,213,214,217,219,221,224,226,228,230,231,232,233],[228,228,227,227,227,227,226,226,226,225,224,223,222,222,221],[257,256,255,254,253,252,252,251,251,251,251,251,251,251,252,253,254,255,257,259,261,264,266,269,271,273,275,276,277,278,278,277,275,273,270,267,264,261,257,255,253,252,251,251,252,254,256,259,262,266,270,274,276],[311,312,313,316,320,325,330,336,340,344,347,349,351,352],[399,399,399,399,398,397,395,392,389,386,383,381,379,377,377,377,378,380,382,385,388,391,393,395,397,398,398,398,398,397,396,395,393,391,389,387,386,384],[420,419,419,419,419,419,419,419,419,419,419,419,419,420,421],[417,417,416,417,419,421,422],[440,440,440,441,441,441,441,441,441,441,441,441,441,442,443,444,446,447,448,449,450,452,453,456,458,460,461,463,464,464,464,464,464,464,465,465,466,467,468],[483,483,484,484,484,484,483,482,481,480,478,477,477,476,476,477,478,480,482,484,487,490,493,496,499,502,504,505,506,505,504],[489,489,490,491,492,492,492,492,492,492,492,492,491,491,491,491,492,493,493,494],[524,523,523,522,522,522,522,522,522,521,521,521,521,521,522,523,524,525,527,529,531,533,535,537,538,540,540,541,541,540,538,537,535,532,529,526,522,519,516,513,512,512,512,514,517,522,527,533,539,542],[588,589,591,594,599,604,609,614,619,623,626,629,630,631],[591,592,593,596,597,601,606,612,616,620,624,626,629,631,633,634,635],[769,770,770,770,769,768,767,765,763,761,759,757,755,754,753,753,753,753,755,757,760,764,768,772,777,780,784,787,790,792,795,796,797,798,798,798,797,797,796,795,795,795,795,796,797,799,801,803,806,808,810,811,812,813,814,814,814,813,812,810,808,806,804,802,801,800,800,800,800,801,803,805,808,812,816,820,825,829,832,835,838,841,843,846,849,851,853,855,855,855,855,855,853,852,851,849,848,847,845,844,843,843,842,843,844,846,848,851,854,856,858,859,860,860,860,859,858,856,853,850,848,846,844,842,841,841],[877,878,878,878,878,879,879,880,880,881,883,885,888,890,892,894,896,897,898,898,898,898,896,894,890,887,884,881,879,879,880,882,884,887,890,893,897,901,905,909,913,916,918,920,921],[939,939,939,939,939,940,940,939,939,938,937,935,934,933,933,933,933,934,936,938,941,943,946,949,952,955,957,960,962,964,965,965,965,962,959,955,951,947,942,938,934,932,932,932,934,936,939,944,949,955,961,967,974,980,982]],"y":[[273,271,270,269,268,268,267,266,266,266,266,267,269,272,275,279,283,287,291,295,298,301,303,305,307,307,308,307,306,304,302,299,296,293,289,286,283,280,278,276,275,275,274,274,276,277,279,282,284,287,290,293,296,298,300,300,300,300,299,298,295,292,289,285,281,277,273,270,267,265,265,264,265,266,268,270,272,274,276,278,279,280,281,281,280,278,277,276,274,273,272,271,269,268,267,266,265,264,264,264,265,266,268,270,272,275,277,280,283,286,289,291,293,294,296,297,298,299,300,301,301,301,299,296,294],[210,209,208,209,211,214,217,221,225,228,231,233,235,235,235,235,235,235,235,235,236,236,236,237,237,238,238,239,239,239],[223,222,222,221,222,223,226,229,233,237,241,245,249,252,253],[264,264,265,268,271,276,279,281,287,289,292,296,298,300,303,304,306,307,308,308,307,305,303,299,295,291,286,282,277,271,267,263,260,258,256,255,255,256,257,260,263,266,269,272,275,277,279,280,281,281,281,280,279],[286,286,286,286,286,285,285,284,283,282,282,282,281,281],[274,271,270,268,267,266,266,266,266,267,268,269,271,272,274,275,277,278,280,282,283,285,287,289,290,292,293,295,296,298,299,300,300,300,300,300,300,298],[273,273,274,276,278,281,286,290,294,297,300,302,303,304,303],[245,244,241,241,241,242,243],[270,271,273,276,280,284,287,290,293,294,295,294,292,290,288,286,283,281,280,278,277,277,276,276,277,278,280,282,285,288,292,295,299,302,304,306,307,308,308],[212,211,211,212,214,216,219,222,226,228,231,233,235,236,237,237,237,237,237,237,237,236,236,235,235,235,235,235,235,235,235],[223,221,218,217,218,220,222,225,229,232,236,239,242,245,248,250,252,254,256,256],[267,263,262,263,265,267,270,274,278,282,287,292,296,301,303,305,306,306,306,305,303,300,297,294,291,287,283,278,273,269,265,261,259,258,256,256,256,257,259,261,264,267,270,273,275,276,277,277,277,277],[274,273,273,273,272,272,272,271,271,270,270,270,269,269],[289,289,288,288,287,286,285,285,284,284,284,283,283,282,281,281,280],[264,261,259,258,257,256,255,255,255,255,256,258,261,264,269,273,279,284,287,290,292,292,293,292,290,289,287,285,283,280,278,275,273,272,271,270,271,273,275,278,281,284,287,290,292,294,295,295,294,292,290,287,283,280,276,272,269,266,264,263,262,261,261,262,263,265,268,2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es una identidad trigonométrica válida.

    Solución:

    Paso 1: En este ejemplo concreto, el LHS parece más complicado. Por tanto, intentaremos simplificar el LHS para llegar al RHS.

    Paso 2: El términocos4θ-sin4θ{"x":[[115,117,117,117,117,116,116,114,113,111,109,106,104,101,98,95,93,92,92,93,94,97,99,102,106,109,113,117,121,125,129,132,136,138,141,143,144,145,145,145,144,143,142,141,140,139,138,138,138,138,138,139,140,142,144,145,147,148,149,150,151,153,154,155,156,156,156,155,154,152,151,149,147,145,144,144,144,146,148,151,155,160,166,172,180,186,192,197,201,204,207,208,209,210,210,210,210,209,208,206,204,203,201,200,199,199,199,200,202,205,208,212,214,217,218,218,217,216,213,211,208,205,203,200,199],[213,213,213,213,214,214,214,213,212,211,210,208,207,206,207,208,209,211,213,214,217,219,221,224,226,228,230,231,232,233],[228,228,227,227,227,227,226,226,226,225,224,223,222,222,221],[257,256,255,254,253,252,252,251,251,251,251,251,251,251,252,253,254,255,257,259,261,264,266,269,271,273,275,276,277,278,278,277,275,273,270,267,264,261,257,255,253,252,251,251,252,254,256,259,262,266,270,274,276],[311,312,313,316,320,325,330,336,340,344,347,349,351,352],[399,399,399,399,398,397,395,392,389,386,383,381,379,377,377,377,378,380,382,385,388,391,393,395,397,398,398,398,398,397,396,395,393,391,389,387,386,384],[420,419,419,419,419,419,419,419,419,419,419,419,419,420,421],[417,417,416,417,419,421,422],[440,440,440,441,441,441,441,441,441,441,441,441,441,442,443,444,446,447,448,449,450,452,453,456,458,460,461,463,464,464,464,464,464,464,465,465,466,467,468],[483,483,484,484,484,484,483,482,481,480,478,477,477,476,476,477,478,480,482,484,487,490,493,496,499,502,504,505,506,505,504],[489,489,490,491,492,492,492,492,492,492,492,492,491,491,491,491,492,493,493,494],[524,523,523,522,522,522,522,522,522,521,521,521,521,521,522,523,524,525,527,529,531,533,535,537,538,540,540,541,541,540,538,537,535,532,529,526,522,519,516,513,512,512,512,514,517,522,527,533,539,542]],"y":[[273,271,270,269,268,268,267,266,266,266,266,267,269,272,275,279,283,287,291,295,298,301,303,305,307,307,308,307,306,304,302,299,296,293,289,286,283,280,278,276,275,275,274,274,276,277,279,282,284,287,290,293,296,298,300,300,300,300,299,298,295,292,289,285,281,277,273,270,267,265,265,264,265,266,268,270,272,274,276,278,279,280,281,281,280,278,277,276,274,273,272,271,269,268,267,266,265,264,264,264,265,266,268,270,272,275,277,280,283,286,289,291,293,294,296,297,298,299,300,301,301,301,299,296,294],[210,209,208,209,211,214,217,221,225,228,231,233,235,235,235,235,235,235,235,235,236,236,236,237,237,238,238,239,239,239],[223,222,222,221,222,223,226,229,233,237,241,245,249,252,253],[264,264,265,268,271,276,279,281,287,289,292,296,298,300,303,304,306,307,308,308,307,305,303,299,295,291,286,282,277,271,267,263,260,258,256,255,255,256,257,260,263,266,269,272,275,277,279,280,281,281,281,280,279],[286,286,286,286,286,285,285,284,283,282,282,282,281,281],[274,271,270,268,267,266,266,266,266,267,268,269,271,272,274,275,277,278,280,282,283,285,287,289,290,292,293,295,296,298,299,300,300,300,300,300,300,298],[273,273,274,276,278,281,286,290,294,297,300,302,303,304,303],[245,244,241,241,241,242,243],[270,271,273,276,280,284,287,290,293,294,295,294,292,290,288,286,283,281,280,278,277,277,276,276,277,278,280,282,285,288,292,295,299,302,304,306,307,308,308],[212,211,211,212,214,216,219,222,226,228,231,233,235,236,237,237,237,237,237,237,237,236,236,235,235,235,235,235,235,235,235],[223,221,218,217,218,220,222,225,229,232,236,239,242,245,248,250,252,254,256,256],[267,263,262,263,265,267,270,274,278,282,287,292,296,301,303,305,306,306,306,305,303,300,297,294,291,287,283,278,273,269,265,261,259,258,256,256,256,257,259,261,264,267,270,273,275,276,277,277,277,277]],"t":[[0,20,21,35,38,49,55,65,72,82,90,100,105,121,122,134,139,149,155,165,172,182,190,200,206,219,222,234,239,249,255,266,272,282,289,300,306,320,322,334,339,349,355,366,382,389,400,406,420,422,434,439,449,455,466,472,482,489,503,514,515,522,533,539,548,555,565,572,582,589,600,605,619,622,633,639,648,655,665,673,682,689,699,705,715,722,733,739,749,755,765,772,782,789,799,805,819,823,833,839,849,856,865,872,882,889,898,905,915,922,931,939,950,955,965,973,982,989,998,1006,1019,1022,1033,1039,1046],[1560,1575,1589,1606,1617,1623,1637,1639,1650,1656,1666,1673,1682,1699,1736,1739,1750,1756,1766,1773,1783,1789,1799,1806,1817,1823,1837,1839,1851,1866],[2074,2086,2091,2107,2124,2137,2139,2151,2156,2166,2173,2182,2189,2200,2205],[2687,2705,2718,2726,2736,2745,2751,2752,2759,2768,2768,2779,2785,2785,2793,2800,2806,2817,2823,2832,2840,2851,2856,2866,2873,2883,2890,2900,2906,2917,2923,2933,2940,2951,2956,2966,2973,2983,2990,3000,3006,3015,3023,3033,3040,3051,3056,3067,3073,3083,3090,3100,3105],[3678,3707,3715,3723,3734,3741,3753,3757,3768,3773,3784,3790,3800,3807],[4279,4291,4296,4307,4317,4324,4334,4340,4351,4357,4367,4374,4384,4390,4401,4407,4417,4423,4434,4440,4451,4457,4467,4474,4484,4490,4501,4507,4517,4523,4534,4540,4551,4557,4567,4574,4584,4590],[4765,4778,4790,4801,4807,4818,4824,4835,4840,4851,4857,4867,4874,4884,4899],[5070,5075,5080,5083,5090,5101,5106],[5407,5441,5451,5457,5468,5474,5485,5491,5502,5507,5518,5549,5557,5567,5574,5585,5591,5601,5607,5618,5624,5634,5641,5651,5657,5668,5674,5685,5691,5701,5707,5718,5724,5732,5741,5751,5757,5768,5773],[6140,6148,6174,6191,6201,6208,6219,6224,6235,6241,6252,6258,6269,6274,6291,6301,6318,6324,6335,6341,6349,6358,6368,6374,6385,6391,6402,6408,6419,6434,6441],[6617,6623,6628,6630,6641,6652,6658,6669,6674,6686,6691,6702,6708,6719,6724,6736,6741,6752,6758,6765],[7171,7181,7191,7216,7225,7235,7241,7250,7258,7269,7275,7286,7292,7302,7308,7319,7325,7336,7341,7352,7358,7369,7375,7386,7391,7402,7408,7419,7425,7436,7441,7452,7458,7469,7475,7486,7491,7502,7508,7519,7525,7536,7541,7552,7558,7569,7575,7586,7593,7599]],"version":"2.0.0"} también puede escribirse como cos2θ2-sin2θ2{"x":[[165,163,142,142,142,143,146,149,153,155,158,163,167,171,174,178,181,183,186,187,187,188,188,188,187,187,186,185,185,184,183,183,183,184,185,188,190,193,196,199,201,203,205,206,206,206,205,204,203,201,199,197,195,193,190,189,188,187,188,190,193,197,202,207,212,217,222,226,231,235,238,241,244,246,247,248,248,248,247,246,244,241,237,234,232,231,231,232,235,238,242,245,248,250,251,251,251,250,247,245,242,239,236,234,232,232],[262,261,261,260,260,260,260,261,261,262,263,264,265,265,266,266,266,266,265,265,263,262,261,260,262,266,270,275,280],[289,289,289,288,288,287,286,285,285,285,286,288,290,291,293,294,295,298,301,303,305,306,306,306,304,301,298,295,292,290,287,286,286,286,288,291,295,300,305,310],[131,131,129,128,127,126,125,123,120,117,113,111,110,110,114,120,128,137,149,155],[330,328,330,333,337,341,344,345,344,341,337,333,330,327,325,324,323],[347,345,345,344,344,344,344,344,345,347,349,351,353,355,355,355,355,353,350,348,345,343,342,342,344,347,351,356,360,363],[406,402,403,404,409,412,419,426,433,440,445,448,449],[538,539,540,540,540,539,538,537,536,533,531,530,530,530,530,532,535,538,542,544,544,542,539,536,533,529,527,525,523,523,523,525,528],[556,556,556,556,556,556,556,556,555,555,555,555,556,557,559,562,565],[562,561,561,562,563],[574,573,573,573,573,573,573,574,574,574,574,574,575,577,579,581,583,585,586,587,588,588,589,589,589,589,589,590,591,592,594,595],[602,602,602,603,603,604,605,606,608,610,611,613,613,614,614,613,612,610,608,606,604,603,605,607,611,616,618],[644,644,644,644,644,644,643,643,642,641,641,640,641,642,644,646,649,652,654,657,659,661,661,660,658,654,651,647,643,641,639,638,638,640,643,648,653,659,664,667],[687,687,687,688,689,691,693,694,694,693,690,686,683,679,678,675,671],[518,519,519,519,518,517,515,513,510,506,501,496,493,494,499,505,514,525],[702,701,699,698,698,697,698,700,703,706,708,710,711,711,709,706,702,699,697,696,698,703,708,715,723,730,736]],"y":[[206,204,231,233,235,236,238,239,239,239,239,238,237,234,233,230,228,225,222,219,217,215,214,213,213,214,216,218,222,225,228,232,235,237,239,240,240,240,238,236,233,231,227,224,221,217,214,211,209,208,207,207,208,210,212,216,219,223,227,230,232,234,234,235,234,233,231,228,225,222,219,215,211,208,205,202,200,199,198,197,197,197,199,201,203,205,208,211,214,217,220,222,225,227,228,229,230,231,232,232,233,233,233,233,232,231],[184,182,181,180,178,177,175,174,173,173,172,172,173,174,176,179,182,184,187,189,191,193,194,194,193,193,191,190,189],[208,207,206,208,211,215,220,226,231,234,236,237,237,237,237,236,235,232,229,224,219,213,207,202,199,197,196,196,198,200,204,208,212,215,218,220,220,220,220,219],[191,190,188,186,185,185,185,187,191,197,206,216,227,237,246,253,259,264,269,270],[188,186,187,190,194,200,208,218,228,237,245,251,256,260,263,265,265],[157,155,154,152,151,150,148,147,146,145,145,145,145,147,149,153,156,160,163,166,168,169,170,169,169,168,167,166,166,165],[209,209,209,209,208,208,208,208,207,207,207,207,206],[206,203,201,200,198,197,196,195,195,195,196,196,197,201,202,204,207,209,212,215,216,218,220,221,221,221,221,221,219,218,216,214,213],[202,201,200,199,198,200,202,204,208,211,214,217,219,221,221,221,219],[180,179,178,180,181],[205,205,204,203,204,205,207,209,211,213,214,215,214,211,208,205,203,201,199,199,199,200,202,205,208,211,214,217,219,221,222,222],[175,174,171,170,169,169,168,168,168,169,170,172,174,176,179,181,184,186,188,190,191,192,193,193,192,191,191],[202,200,198,199,201,203,205,209,212,216,221,224,227,229,229,228,227,224,221,216,211,206,200,196,192,190,190,190,190,192,194,197,200,203,205,207,208,209,209,209],[189,188,187,187,189,192,197,203,211,220,228,235,241,246,248,251,253],[184,181,180,179,178,178,178,178,181,186,194,205,217,228,237,245,250,252],[151,151,146,145,143,142,142,143,144,145,147,150,152,156,159,162,165,167,168,169,168,168,167,166,165,164,164]],"t":[[0,18,32,42,43,43,51,61,68,76,77,85,93,102,106,113,123,130,140,146,156,163,173,180,190,213,223,230,240,246,256,263,273,280,290,296,306,313,323,330,340,346,356,363,373,380,390,396,406,413,423,430,440,446,456,463,473,480,490,496,506,513,523,530,540,546,557,563,573,580,590,597,606,613,621,630,640,646,657,663,673,680,690,696,707,713,723,730,740,747,757,763,773,780,790,796,807,813,823,830,840,847,857,863,873,879],[1169,1176,1181,1188,1197,1207,1213,1224,1230,1241,1247,1257,1263,1274,1280,1291,1297,1307,1313,1324,1330,1341,1347,1357,1388,1397,1407,1413,1424],[1837,1852,1864,1889,1897,1907,1914,1924,1930,1941,1947,1958,1964,1969,1977,1978,1985,1991,1997,2008,2014,2025,2030,2041,2047,2058,2064,2074,2080,2091,2097,2107,2114,2124,2130,2141,2147,2157,2164,2174],[2789,2794,2800,2802,2807,2814,2824,2831,2842,2847,2858,2864,2874,2881,2891,2897,2908,2914,2924,2929],[3338,3349,3381,3391,3398,3409,3414,3425,3431,3442,3448,3458,3464,3475,3481,3492,3497],[3848,3861,3867,3873,3881,3893,3898,3909,3914,3925,3931,3942,3948,3959,3964,3975,3981,3992,3998,4008,4014,4025,4031,4048,4056,4064,4075,4081,4092,4097],[4454,4467,4487,4487,4495,4498,4509,4515,4526,4531,4542,4548,4559],[5151,5168,5174,5182,5192,5198,5211,5215,5226,5232,5243,5250,5250,5260,5265,5276,5282,5292,5298,5309,5315,5326,5332,5342,5348,5359,5365,5376,5382,5392,5398,5409,5415],[5535,5541,5547,5549,5565,5607,5615,5626,5632,5643,5648,5660,5665,5676,5682,5693,5698],[5806,5816,5822,5849,5857],[6008,6019,6024,6048,6073,6082,6093,6099,6110,6115,6127,6132,6157,6165,6177,6182,6193,6198,6209,6215,6226,6232,6243,6249,6260,6265,6276,6282,6293,6299,6310,6315],[6476,6482,6488,6491,6499,6507,6515,6526,6532,6544,6549,6560,6565,6577,6582,6594,6599,6610,6615,6627,6632,6644,6660,6665,6676,6682,6689],[7042,7057,7065,7096,7104,7113,7114,7121,7127,7132,7141,7149,7160,7166,7177,7182,7194,7199,7211,7216,7227,7232,7244,7249,7261,7266,7278,7282,7294,7299,7311,7316,7327,7332,7344,7349,7361,7366,7378,7381],[7537,7549,7574,7582,7594,7599,7611,7616,7628,7632,7644,7649,7661,7666,7672,7680,7688],[8155,8167,8175,8183,8194,8199,8211,8216,8228,8233,8245,8249,8261,8266,8278,8283,8295,8299],[8681,8692,8700,8702,8709,8716,8741,8750,8761,8766,8778,8783,8795,8800,8812,8816,8828,8833,8845,8850,8866,8878,8883,8895,8900,8912,8916]],"version":"2.0.0"} y ahora se puede escribir como producto de dos términos utilizando la identidad algebraica a2-b2=a+ba-b{"x":[[172,172,172,172,173,173,173,172,170,167,164,161,156,151,147,143,139,136,135,134,134,135,137,139,141,143,145,148,150,152,154,157,159,162,164,166,167,168,169,170,171,172,173,174,175,176,177,178,179,181,183,186],[184,183,182,182,182,182,184,185,187,189,191,192,194,195,196,196,196,196,195,194,192,190,188,186,186,186,188,190,194,199,204,209,214,219,221],[236,236,237,239,244,250,256,263,269,273,277,280,281,282],[336,337,337,337,338,338,338,338,338,338,338,338,337,337,337,337,337,337,337,337,338,339,340,341,342,344,346,348,351,353,356,358,360,362,363,364,364,365,365,364,363,361,358,353,348,343,339,335,333,333,334],[371,371,371,371,372,374,376,378,380,381,382,383,383,383,382,381,379,378,376,375,374,373,373,374,375,378,382,387,392],[438,436,435,437,439,443,447,451,455,459,462,465,467],[431,430,428,427,428,429,433,438,443,448,451],[570,572,575,576],[581,581,581,581,580,580,579,578,577,575,573,570,567,564,561,558,555,554,553,553,554,556,558,560,563,566,569,572,575,578,581,583,585,586,587,587,587,587,587,588,588,589,591,592,594,596,598,602],[618,617,619,622,627,632,636,641,644,646,647],[630,630,628,628,629,629,630,630,630,630,630,630,630,630,630,631,632],[677,676,676,675,675,675,675,675,674,673,672,671,671,670,670,670,670,671,672,674,677,679,682,684,686,688,690,691,692,693,693,692,690,688,684,681,677,674,672,671,671],[739,739,739,739,739,739,739,739,738,735,732,728,724,719,714,709,705,702,701],[533,532,532,532,532,532,532,530,528,525,520,517,514,513,513,516,520,523],[812,811,808,806,802,797,792,788,784,780,778,777,776,777,779,782,785,790,795],[847,847,846,845,845,845,845,844,843,842,841,840,837,835,832,829,827,825,825,825,826,828,830,832,835,838,841,844,847,851,854,857,859,862,864,866,868,870,871,873,875,877,879,880],[898,898,898,899,900,903,907,912,916,920],[955,955,955,954,953,952,951,950,948,946,945,944,943,942,942,942,942,942,943,944,946,949,951,954,957,959,961,963,964,965,965,965,963,960,957,953,949,945,943,941,941],[991,992,993,995,997,999,1001,1003,1004,1003,1001,997,993,988,983,977,971,963,958,957]],"y":[[269,268,266,263,261,258,255,253,250,248,247,247,248,250,254,259,265,272,278,283,286,289,290,290,291,290,289,286,284,280,276,272,267,263,259,258,257,258,261,264,268,272,275,278,282,284,286,287,287,287,286,283],[213,211,207,205,203,202,200,198,197,195,194,194,195,196,198,201,204,207,211,215,219,223,227,230,232,234,234,234,234,232,230,229,227,226,225],[268,269,269,269,269,268,266,264,263,262,261,260,260,260],[217,216,213,212,213,214,217,220,224,229,234,240,246,252,258,263,268,272,275,277,279,280,280,280,278,276,273,270,267,265,264,262,262,263,264,266,269,272,276,279,283,285,288,289,291,291,291,290,287,283,277],[192,191,188,186,185,183,181,179,178,177,177,177,178,180,183,186,189,191,194,196,198,200,201,201,201,201,201,200,198],[250,250,250,249,248,248,247,247,247,247,247,247,247],[272,272,272,272,272,272,271,271,270,269,269],[247,247,247,247],[244,243,241,239,239,238,237,236,235,235,235,235,235,237,239,242,246,249,252,255,256,258,259,259,259,259,258,256,254,251,248,245,242,240,240,242,245,249,253,257,261,265,268,270,271,271,271,269],[248,248,248,248,248,248,248,248,247,247,247],[237,236,235,234,236,238,242,245,249,254,259,263,267,269,270,271,271],[208,208,207,206,207,210,215,220,226,233,239,245,249,252,255,256,257,256,254,252,249,247,245,243,242,242,244,246,249,252,255,257,260,263,265,267,269,269,269,268,267],[207,208,210,213,218,225,233,241,250,256,262,268,273,278,282,285,288,289,289],[218,216,214,213,212,211,210,211,214,220,229,239,251,263,272,281,288,291],[209,206,204,204,207,213,220,229,239,249,258,267,274,279,284,287,290,291,290],[250,249,247,245,244,243,240,238,236,234,233,233,234,236,240,244,248,253,257,259,261,263,263,263,262,260,257,254,251,249,246,245,245,245,246,248,250,252,254,256,258,259,260,260],[246,245,244,243,243,243,243,243,243,243],[195,194,192,193,195,199,205,211,219,227,234,240,245,249,252,253,254,255,255,253,252,249,247,246,245,244,244,245,247,250,253,256,259,261,263,265,267,268,268,268,266],[202,201,199,199,203,208,215,223,233,242,251,260,266,273,278,283,287,290,293,293]],"t":[[0,10,19,22,30,39,47,58,63,75,80,91,97,108,113,124,130,141,147,157,163,174,180,191,197,207,213,224,230,241,247,258,263,274,280,291,297,308,313,324,330,341,347,357,363,374,380,391,397,408,413,424],[700,707,712,714,722,730,740,747,757,764,774,780,791,797,807,814,824,830,840,847,857,864,874,880,894,897,907,914,924,930,940,947,957,964,972],[1262,1269,1283,1291,1297,1308,1314,1324,1331,1341,1347,1357,1364,1374],[1608,1615,1620,1631,1647,1657,1664,1674,1681,1691,1697,1708,1714,1724,1731,1741,1747,1758,1764,1774,1781,1797,1808,1814,1824,1831,1841,1847,1858,1864,1874,1881,1891,1897,1908,1914,1926,1931,1941,1947,1956,1964,1975,1981,1991,1997,2008,2014,2024,2031,2041],[2408,2416,2422,2424,2431,2440,2448,2458,2464,2473,2481,2491,2498,2508,2514,2525,2531,2541,2548,2558,2564,2575,2581,2591,2598,2608,2614,2625,2631],[3100,3116,3123,3139,3148,3159,3165,3175,3181,3192,3198,3208,3214],[3362,3370,3377,3380,3386,3389,3398,3409,3415,3423,3431],[3699,3703,3710,3712],[3911,3922,3929,3931,3937,3940,3948,3959,3965,3976,3982,3992,3998,4009,4015,4025,4032,4042,4048,4059,4065,4076,4081,4092,4098,4109,4115,4125,4132,4142,4148,4159,4165,4176,4182,4207,4215,4225,4232,4242,4248,4259,4265,4276,4282,4292,4298,4309],[4567,4581,4607,4615,4626,4632,4643,4648,4659,4665,4676],[4824,4834,4841,4844,4857,4865,4876,4882,4893,4899,4910,4915,4927,4932,4943,4948,4956],[5122,5130,5137,5140,5157,5165,5174,5182,5193,5199,5209,5215,5226,5232,5243,5249,5260,5290,5299,5309,5315,5326,5332,5343,5349,5360,5365,5376,5382,5393,5399,5410,5415,5426,5432,5443,5449,5460,5465,5476,5481],[5676,5684,5694,5702,5709,5715,5726,5732,5743,5749,5760,5765,5777,5782,5793,5800,5810,5815,5823],[6238,6247,6253,6255,6261,6274,6291,6309,6316,6327,6332,6343,6349,6360,6366,6377,6382,6390],[6852,6859,6865,6867,6874,6883,6894,6899,6910,6916,6927,6933,6944,6949,6961,6966,6977,6983,6994],[7257,7268,7275,7278,7285,7293,7299,7311,7316,7327,7333,7344,7349,7361,7366,7377,7383,7394,7400,7411,7416,7427,7433,7444,7449,7461,7466,7477,7483,7494,7500,7511,7516,7527,7533,7544,7550,7561,7566,7578,7583,7594,7599,7607],[7750,7757,7764,7767,7775,7783,7795,7800,7812,7816],[8084,8093,8100,8103,8111,8116,8128,8133,8145,8150,8162,8166,8179,8183,8195,8200,8212,8216,8233,8244,8250,8261,8266,8278,8283,8295,8300,8311,8316,8328,8333,8345,8350,8361,8367,8378,8383,8395,8400,8411,8416],[8676,8685,8690,8693,8700,8708,8717,8727,8733,8745,8750,8761,8767,8778,8783,8795,8800,8811,8817,8824]],"version":"2.0.0"}. Por tanto, tenemos:

    cos2θ2-sin2θ2=cos2θ+sin2θcos2θ-sin2θ

    Paso 3 : Ahora podemos aplicar la identidad pitagórica, cos2θ+sin2θ=1{"x":[[107,107,106,105,105,104,103,102,100,99,97,95,93,90,88,86,85,84,85,86,88,91,94,97,101,104,108,111,114,117,120,122,124,125,126,128,128,129,129,129,129,128,128,128,128,129,129,130,130,131,132,133,134,135,136,138,140,142,144,148,150,151,152,152,152,151,150,148,146,144,141,139,137,135,134,134,134,135,136,139,142,146,151,156,162,167,173,178,183,188,193,197,200,204,206,208,209,209,208,207,206,204,202,200,198,196,194,193,193,193,193,195,198,201,204,207,209,211,211,211,211,210,209,207,205,204,202],[225,223,221,220,219,218,218,219,221,222,224,227,229,231,233,234,236,237,238,238,238,238,237,236,235,235,234,235,237,240,245,249,254,259,261],[265,264,261,260,260,259,259,259,259,259,261,262,264,270,272,275,278,281,283,286,287,289,290,290,290,289,287,285,282,279,275,272,269,268,267,268,270,273,277,282,287],[359,358,358,357,357,356,356,356,356,356,356,356,356],[335,334,332,334,337,339,341,346,349,355,361,367,371,375,378,380,382,383,384,385,386,386],[445,445,445,445,445,445,444,443,441,439,436,433,430,427,424,422,421,421,421,423,425,428,431,433,435,437,439,440,441,441,440,438,436,433,431],[461,461,461,462,462,462,463,463,462,461,459,459],[467,467,468,468],[487,487,487,487,487,488,488,488,489,489,489,490,490,491,492,492,492,492,492,492,493,493,495,497,499,502,505,506,507,508,508,509,510,510,511,512,513,514,515,516,518,520,522,524,527],[524,524,524,524,524,525,527,528,531,533,535,537,538,540,541,541,542,542,542,542,541,539,537,534,532,531,530,530,531,533,535,538,542,547,552,556,561,564],[598,597,596,595,594,593,592,592,592,592,592,593,595,596,598,600,602,604,606,608,609,611,613,614,615,616,617,617,617,615,613,610,607,603,599,595,592,590,589,589,591,594,598,604,609,616],[666,665,665,666,668,670,672,674,677,679,682,685,688,691,693,694],[663,663,663,663,665,667,671,677,682,688,692,695,698,699,700],[748,749,749,749,750,750,750,750,750,749,748,747,746,745,745,746]],"y":[[258,257,255,254,253,253,252,252,252,251,251,252,254,257,261,266,271,276,280,284,288,290,292,293,294,294,293,293,291,290,289,287,285,283,281,277,275,272,270,269,268,268,269,271,273,275,278,281,284,288,291,294,296,297,298,298,297,296,293,287,283,278,274,270,267,264,262,261,260,260,260,261,262,263,264,266,268,270,273,276,278,280,281,282,282,282,281,279,276,273,269,265,261,257,254,251,248,247,246,245,245,245,245,246,248,250,251,253,255,257,259,262,265,268,270,273,275,277,278,279,280,281,281,282,282,281,280],[216,214,212,212,211,210,209,207,205,203,202,200,199,198,198,198,199,200,202,204,207,210,213,217,220,223,225,227,227,227,226,224,223,221,221],[252,251,249,249,251,254,258,262,266,270,274,277,280,282,282,282,281,279,276,273,269,264,259,254,249,246,244,242,242,242,243,246,248,251,254,256,258,259,259,259,259],[251,251,254,259,267,271,280,288,295,301,306,310,313],[273,272,271,271,271,271,271,271,271,271,270,269,269,268,268,268,268,268,268,268,268,269],[264,262,261,260,259,258,256,255,255,254,254,255,256,258,260,262,264,266,268,270,271,273,275,276,278,280,281,283,284,285,286,287,288,289,290],[263,262,261,263,267,272,277,283,289,293,297,298],[241,238,238,239],[276,275,274,273,272,272,274,276,279,283,288,291,294,297,298,297,296,294,292,290,288,286,284,281,279,276,275,274,274,274,275,277,279,282,285,289,292,294,297,298,300,300,300,300,299],[229,228,224,223,222,222,222,221,221,220,220,220,220,220,221,222,224,226,228,231,233,237,240,244,247,250,251,252,251,251,250,249,247,246,245,244,244,244],[257,256,255,256,258,261,265,270,275,280,285,289,292,294,294,295,295,294,293,291,289,285,281,276,270,264,258,253,248,246,244,244,244,245,248,251,254,258,261,264,266,268,269,269,269,268],[260,259,258,258,258,259,259,259,260,260,260,260,260,260,260,260],[280,279,278,277,277,277,276,276,276,276,275,275,275,275,275],[236,238,240,246,256,266,274,282,288,294,299,303,307,310,312,313]],"t":[[0,10,17,20,29,36,43,52,59,69,76,86,93,102,109,119,126,136,143,152,159,169,177,186,193,202,209,219,226,236,243,252,259,271,276,287,293,306,309,319,326,336,372,376,386,393,403,409,419,426,438,443,453,459,469,477,486,493,503,522,526,534,543,552,559,569,583,585,593,602,609,619,626,635,643,652,659,669,676,684,693,702,710,719,726,735,743,752,759,769,777,785,793,802,809,819,826,835,843,852,859,869,876,885,893,902,909,919,926,935,943,952,959,969,976,985,993,1002,1010,1019,1026,1036,1043,1052,1059,1069,1078],[1404,1414,1419,1426,1438,1452,1460,1469,1476,1485,1493,1502,1510,1520,1530,1535,1543,1554,1560,1570,1576,1586,1593,1603,1610,1620,1626,1647,1654,1660,1671,1676,1686,1693,1700],[2130,2143,2148,2157,2160,2170,2177,2187,2193,2204,2210,2220,2227,2243,2255,2260,2273,2277,2286,2293,2303,2310,2319,2327,2336,2343,2353,2360,2370,2377,2386,2393,2403,2411,2424,2427,2436,2443,2455,2460,2471],[4710,4723,4739,4748,4756,4761,4771,4778,4788,4794,4805,4811,4821],[5079,5086,5094,5106,5114,5123,5123,5132,5138,5144,5155,5161,5171,5178,5188,5194,5205,5211,5221,5228,5238,5243],[5614,5619,5624,5627,5632,5638,5645,5655,5661,5672,5678,5688,5695,5705,5711,5722,5728,5738,5745,5755,5761,5772,5778,5788,5795,5805,5811,582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    cos4θ-sin4θ=cos2θ-sin2θ{"x":[[166,165,164,163,162,161,160],[164,164,164,164,164,164,163,163,162,161,159,156,152,149,146,143,141,140,140,141,143,146,149,153,158,163,168,172,177,179,182,182,182,182,181,181,179,178,177,177,177,178,179,180,182,184,187,190,193,195,197,199,200,201,201,201,200,199,197,194,191,188,186,184,184,183,184,186,189,192,196,201,205,210,214,217,220,222,223,224,224,224,223,222,221,220,219,219,219,220,221,223,225,227,229,230,231,231,231,229,227,226,224,222,221],[242,242,243,243,243,243,242,241,240,239,239,239,238,238,238,238,237,237,238,239,241,243,246,249,253,256,258,260,261,262],[253,253,252,252,252,252,250,250,247,248,249,250,251,252,253],[278,278,278,278,277,277,277,276,275,274,273,272,271,271,272,273,276,279,282,285,288,290,292,293,294,294,292,290,287,284,281,278,276,274,274,275,277,281,285,291,296,299],[325,324,322,321,322,324,327,331,336,340,343,346,348],[389,389,390,390,390,390,390,390,389,388,386,385,383,382,381,381,382,383,385,387,389,389,390,389,387,384,382,379,377,375,374,374],[406,406,406,406,406,406,405,404,404,403,403,403,403,403,404,406,409,411],[409,409,410,411,412],[420,420,420,421,421,421,421,421,421,421,421,421,421,422,424,426,429,432,434,436,438,438,439,439,439,438,438,438,439,440,441,443],[459,459,459,459,459,459,458,458,457,456,455,455,455,455,456,458,461,464,468,471,474,475,476],[468,468,468,468,468,469,469,470,470,470,470,469,469,468,467,467,467,467],[504,503,503,502,502,502,502,501,501,500,500,500,501,502,504,506,509,512,514,517,519,521,522,523,523,521,518,515,511,508,505,503,501,501,501,503,506,511,516,522,529,532],[570,569,566,567,568,571,575,579,582,587,590],[566,565,564,566,569,573,578,584,590,596,600,602],[666,665,664,663,662,660,657,654,651,649,647,646,646,646,648,651,654,658,662,666,670,674,677,680,681,682,682,682,682,681,680,680,680,680,680,680,681,682,683,684,686,688,689,691,693,694,696,697,697,697,696,694,693,691,690,688,687,687,687,687,689,691,695,699,703,707,711,714,717,719,721,722,722,721,720,718,716,714,713,713,713,715,717,719,722,725,726,727,727,725,722,720,717,714,712,710,710],[735,735,734,734,734,734,734,735,736,738,741,744,747,749,750,751,751,750,749,747,745,743,741,741,740,741,743,747,751,757,760],[770,770,770,770,770,769,769,768,767,766,765,765,765,766,767,770,772,776,779,783,787,790,792,792,792,789,785,781,777,773,770,768,766,766,767,769,773,778,783,788],[818,817,819,820,824,828,832,836,839,840,841],[890,890,890,891,891,891,890,889,887,885,882,879,876,875,874,875,876,878,881,884,887,888,889,888,886,883,880,877,876,874,874],[902,902,902,901,901,900,899,899,899,899,898,898,898,898,897,897,898],[903,903,903,903,904],[915,914,914,914,914,914,914,914,914,913,913,913,913,913,913,913,914,916,918,921,924,926,929,931,932,932,933,933,934,934,934,934,935,935,936,938],[956,955,955,954,954,954,955,956,958,960,962,965,967,969,970,971,971,971,970,968,966,964,963,962,962,964,967,971,975],[992,993,993,993,994,994,994,994,993,992,991,990,989,988,988,988,989,992,994,998,1001,1004,1008,1010,1013,1013,1013,1011,1007,1004,1000,997,994,992,991,992,994,996,1000,1005,1010,1015]],"y":[[219,219,219,219,219,219,219],[219,218,219,218,217,216,216,215,215,215,215,215,217,219,222,227,231,236,241,245,248,250,251,251,251,249,247,243,239,235,232,229,227,225,224,223,223,223,225,227,230,233,235,238,241,243,245,245,245,244,242,240,237,233,229,226,222,219,217,216,216,217,218,221,225,228,231,234,237,238,239,239,239,237,235,232,229,226,223,220,217,215,214,213,212,212,212,213,215,217,219,222,225,228,232,236,239,242,244,245,246,246,246,245,243],[179,178,178,177,178,179,180,182,184,186,187,189,190,192,193,194,195,196,197,197,197,196,195,194,193,192,192,191,191,191],[187,186,186,185,184,186,187,190,193,196,200,203,205,207,208],[224,223,222,220,219,218,217,219,222,226,230,234,239,242,246,248,249,249,247,245,241,237,232,227,220,215,210,206,204,204,205,207,210,215,219,222,226,228,229,230,229,229],[227,227,227,227,227,227,227,227,227,227,227,226,226],[227,226,224,222,221,220,218,217,217,216,216,217,218,220,222,225,227,230,232,233,235,236,237,238,238,238,238,238,237,236,235,234],[227,225,224,222,221,220,220,222,224,227,230,233,236,238,240,241,241,240],[205,203,203,204,205],[224,223,222,221,222,223,225,227,229,230,231,232,233,232,230,227,224,221,219,218,217,218,221,224,227,231,234,237,239,241,241,241],[186,185,184,183,182,183,185,187,190,192,195,197,199,200,201,201,201,201,200,199,198,198,197],[195,194,193,192,191,191,190,190,191,192,194,197,200,203,206,210,212,214],[214,212,211,210,209,211,214,218,222,226,230,234,237,239,241,241,241,240,237,234,229,224,218,212,207,203,201,200,200,201,204,207,210,214,217,219,222,223,224,225,224,224],[215,214,213,213,213,213,213,213,213,213,214],[232,232,231,231,230,229,228,228,228,228,228,229],[216,214,213,212,211,211,212,213,216,220,224,227,231,234,236,238,238,238,238,237,235,233,231,229,226,224,223,221,220,220,221,223,226,229,231,235,237,240,242,242,242,241,239,236,233,229,225,221,218,214,213,212,212,212,214,216,219,222,225,228,230,232,233,233,233,231,229,227,2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    Paso 4 : Aplicando la fórmula del semiángulo cos2θ=cos2θ-sin2θ{"x":[[96,97,97,97,98,98,99,99,100,100,100,100,99,98,95,93,90,87,85,83,81,79,78,77,77,77,78,80,82,85,87,91,95,99,102,106,109,112,115,117,119,121,122,122,122,122,121,120,119,118,117,116,116,115,115,115,115,115,115,116,117,119,120,122,124,125,127,129,131,132,134,135,136,136,136,136,135,133,131,130,128,126,125,124,123,123,123,123,123,124,127,130,133,138,144,149,156,161,167,172,177,181,185,188,189,190,190,190,189,188,185,182,179,176,172,169,167,164,163,163,163,164,166,169,172,176,180,184,187,190,191,192,192,192,191,189,186,183,179,176,174],[225,224,223,222,222,222,223,225,228,232,236,241,245,249,252,254,255,255,254,252,250,247,243,239,235,231,228,225,223,222,221,221,223,225,228,231,236,240,245,250,256,260,264,267,269,271,271],[300,300,299,299,299,298,296,295,293,291,290,290,290,292,294,297,300,304,308,312,317,320,324,326,328,328,328,327,326,323,320,317,313,310,306,303,301,299,298,298,299,301,304,308,312,317,323,326],[370,369,368,367,368,369,371,374,377,381,385,389,393,397,399,400],[363,362,363,364,366,369,374,380,386,392,398,404,408,411,414,415,416],[461,461,462,463,463,464,464,463,462,460,457,454,451,448,445,443,441,441,441,443,445,448,451,455,459,463,467,470,474,477,479,481,482,483,483,483,482,481,480,479,478,477,476,475,475,474,474,474,475,476,477,479,481,484,486,489,491,494,496,497,499,499,499,499,497,496,494,492,490,488,486,484,483,482,482,482,484,487,490,494,499,505,511,517,523,529,535,540,544,548,550,551,551,551,550,548,545,542,539,536,534,532,531,531,531,532,535,538,541,545,548,550,552,553,553,553,552,550,548,547,545,543,542],[570,570,569,569,569,570,572,574,576,579,581,584,587,589,590,591,591,591,589,587,585,582,580,578,577,577,578,580,584,588,592,594],[607,607,607,608,608,608,608,607,606,604,602,600,599,598,597,597,597,598,600,602,605,608,612,616,619,623,625,627,628,628,627,626,623,620,617,614,611,607,604,602,600,600,600,603,606,612,618,625,632,640],[673,673,673,675,677,680,684,688,692,695,698,700,701,700],[758,759,760,763,764,766,766,767,765,763,759,755,750,746,741,738,735,734,734,734,737,740,743,746,748,751,752,753,754,754,753,752,750,748,746,744,742,741,740,739],[793,792,791,790,789,787,785,783,782,780,779,779],[791,791,792,792],[805,805,807,808,808,808,808,808,808,808,807,807,807,807,807,807,807,807,808,810,811,813,816,818,820,822,824,825,826,827,828,829,830,830,831,831,832,832,833,834,836,838,840],[851,851,852,852,853,854,855,856,857,858,859,860,862,865,867,870,873,875,876,877,877,876,874,871,867,864,861,860,860,863,867,871,877,882,884],[905,906,907,908,908,907,905,903,901,900,898,898,898,898,898,900,901,903,906,909,912,914,917,919,921,923,924,925,925,924,922,920,918,915,911,908,904,901,898,896,896,896,899,904,910,918,927,936]],"y":[[289,289,288,287,286,285,284,282,281,279,278,277,276,275,275,274,275,276,278,280,283,287,291,295,300,305,309,312,315,317,319,320,320,320,320,318,317,315,312,310,307,303,300,297,295,294,293,293,293,293,294,295,296,298,300,303,305,309,312,316,320,323,325,327,327,327,326,324,322,319,316,312,308,304,299,295,291,289,287,286,285,285,285,287,289,291,294,298,301,304,307,309,310,310,310,309,307,305,302,298,295,292,288,285,282,279,277,275,273,272,271,271,271,272,273,275,276,279,281,283,285,288,290,292,294,296,299,301,303,305,307,309,311,313,314,316,317,318,319,319,319],[288,288,287,285,283,280,278,275,272,270,268,266,265,265,266,268,271,275,280,286,291,296,301,306,310,314,317,320,323,325,327,328,329,329,329,329,327,326,324,323,321,320,319,318,318,318,319],[278,277,277,278,280,284,289,295,301,307,312,316,320,322,323,324,324,323,320,316,312,306,299,293,286,281,276,273,270,268,267,267,267,269,272,276,280,285,289,293,295,297,299,299,299,299,297,296],[285,284,284,283,283,283,283,284,284,284,283,282,282,281,281,281],[306,305,303,303,303,303,303,303,302,301,300,298,296,295,294,293,293],[286,285,283,281,280,280,279,279,279,281,283,285,289,293,297,302,306,310,313,316,318,320,321,321,321,321,319,317,315,313,310,308,305,302,300,298,296,295,294,294,294,295,297,299,302,305,309,312,316,320,323,325,326,326,326,324,322,319,315,311,307,303,298,294,291,289,287,287,287,287,289,292,295,299,302,306,309,312,313,315,315,315,313,310,308,304,301,297,293,290,287,284,282,280,279,278,278,278,278,279,281,282,284,286,288,291,293,296,298,301,303,306,308,310,311,313,314,315,316,316,316,316,315],[241,240,236,235,233,231,230,228,226,225,224,224,224,225,226,228,230,233,236,240,243,247,249,252,253,254,254,254,254,254,254,253],[278,277,274,273,272,273,274,277,280,285,289,294,299,305,310,314,318,320,321,322,322,320,318,314,310,305,299,294,288,282,277,272,269,267,266,266,267,269,273,277,282,287,290,294,296,297,297,296,294,291],[292,293,294,294,294,293,293,292,291,290,289,289,288,288],[284,283,281,278,277,275,273,272,270,269,268,268,268,269,271,273,276,278,280,283,285,287,289,290,292,294,296,298,300,301,303,304,306,307,307,307,307,307,305,304],[277,276,278,281,284,289,293,297,302,305,307,309],[245,242,242,243],[285,284,280,280,281,283,286,288,291,293,295,297,299,300,301,302,301,299,296,293,290,287,28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    cos4θ-sin4θ=cos2θ{"x":[[115,117,117,117,117,116,116,114,113,111,109,106,104,101,98,95,93,92,92,93,94,97,99,102,106,109,113,117,121,125,129,132,136,138,141,143,144,145,145,145,144,143,142,141,140,139,138,138,138,138,138,139,140,142,144,145,147,148,149,150,151,153,154,155,156,156,156,155,154,152,151,149,147,145,144,144,144,146,148,151,155,160,166,172,180,186,192,197,201,204,207,208,209,210,210,210,210,209,208,206,204,203,201,200,199,199,199,200,202,205,208,212,214,217,218,218,217,216,213,211,208,205,203,200,199],[213,213,213,213,214,214,214,213,212,211,210,208,207,206,207,208,209,211,213,214,217,219,221,224,226,228,230,231,232,233],[228,228,227,227,227,227,226,226,226,225,224,223,222,222,221],[257,256,255,254,253,252,252,251,251,251,251,251,251,251,252,253,254,255,257,259,261,264,266,269,271,273,275,276,277,278,278,277,275,273,270,267,264,261,257,255,253,252,251,251,252,254,256,259,262,266,270,274,276],[311,312,313,316,320,325,330,336,340,344,347,349,351,352],[399,399,399,399,398,397,395,392,389,386,383,381,379,377,377,377,378,380,382,385,388,391,393,395,397,398,398,398,398,397,396,395,393,391,389,387,386,384],[420,419,419,419,419,419,419,419,419,419,419,419,419,420,421],[417,417,416,417,419,421,422],[440,440,440,441,441,441,441,441,441,441,441,441,441,442,443,444,446,447,448,449,450,452,453,456,458,460,461,463,464,464,464,464,464,464,465,465,466,467,468],[483,483,484,484,484,484,483,482,481,480,478,477,477,476,476,477,478,480,482,484,487,490,493,496,499,502,504,505,506,505,504],[489,489,490,491,492,492,492,492,492,492,492,492,491,491,491,491,492,493,493,494],[524,523,523,522,522,522,522,522,522,521,521,521,521,521,522,523,524,525,527,529,531,533,535,537,538,540,540,541,541,540,538,537,535,532,529,526,522,519,516,513,512,512,512,514,517,522,527,533,539,542],[588,589,591,594,599,604,609,614,619,623,626,629,630,631],[591,592,593,596,597,601,606,612,616,620,624,626,629,631,633,634,635],[769,770,770,770,769,768,767,765,763,761,759,757,755,754,753,753,753,753,755,757,760,764,768,772,777,780,784,787,790,792,795,796,797,798,798,798,797,797,796,795,795,795,795,796,797,799,801,803,806,808,810,811,812,813,814,814,814,813,812,810,808,806,804,802,801,800,800,800,800,801,803,805,808,812,816,820,825,829,832,835,838,841,843,846,849,851,853,855,855,855,855,855,853,852,851,849,848,847,845,844,843,843,842,843,844,846,848,851,854,856,858,859,860,860,860,859,858,856,853,850,848,846,844,842,841,841],[877,878,878,878,878,879,879,880,880,881,883,885,888,890,892,894,896,897,898,898,898,898,896,894,890,887,884,881,879,879,880,882,884,887,890,893,897,901,905,909,913,916,918,920,921],[939,939,939,939,939,940,940,939,939,938,937,935,934,933,933,933,933,934,936,938,941,943,946,949,952,955,957,960,962,964,965,965,965,962,959,955,951,947,942,938,934,932,932,932,934,936,939,944,949,955,961,967,974,980,982]],"y":[[273,271,270,269,268,268,267,266,266,266,266,267,269,272,275,279,283,287,291,295,298,301,303,305,307,307,308,307,306,304,302,299,296,293,289,286,283,280,278,276,275,275,274,274,276,277,279,282,284,287,290,293,296,298,300,300,300,300,299,298,295,292,289,285,281,277,273,270,267,265,265,264,265,266,268,270,272,274,276,278,279,280,281,281,280,278,277,276,274,273,272,271,269,268,267,266,265,264,264,264,265,266,268,270,272,275,277,280,283,286,289,291,293,294,296,297,298,299,300,301,301,301,299,296,294],[210,209,208,209,211,214,217,221,225,228,231,233,235,235,235,235,235,235,235,235,236,236,236,237,237,238,238,239,239,239],[223,222,222,221,222,223,226,229,233,237,241,245,249,252,253],[264,264,265,268,271,276,279,281,287,289,292,296,298,300,303,304,306,307,308,308,307,305,303,299,295,291,286,282,277,271,267,263,260,258,256,255,255,256,257,260,263,266,269,272,275,277,279,280,281,281,281,280,279],[286,286,286,286,286,285,285,284,283,282,282,282,281,281],[274,271,270,268,267,266,266,266,266,267,268,269,271,272,274,275,277,278,280,282,283,285,287,289,290,292,293,295,296,298,299,300,300,300,300,300,300,298],[273,273,274,276,278,281,286,290,294,297,300,302,303,304,303],[245,244,241,241,241,242,243],[270,271,273,276,280,284,287,290,293,294,295,294,292,290,288,286,283,281,280,278,277,277,276,276,277,278,280,282,285,288,292,295,299,302,304,306,307,308,308],[212,211,211,212,214,216,219,222,226,228,231,233,235,236,237,237,237,237,237,237,237,236,236,235,235,235,235,235,235,235,235],[223,221,218,217,218,220,222,225,229,232,236,239,242,245,248,250,252,254,256,256],[267,263,262,263,265,267,270,274,278,282,287,292,296,301,303,305,306,306,306,305,303,300,297,294,291,287,283,278,273,269,265,261,259,258,256,256,256,257,259,261,264,267,270,273,275,276,277,277,277,277],[274,273,273,273,272,272,272,271,271,270,270,270,269,269],[289,289,288,288,287,286,285,285,284,284,284,283,283,282,281,281,280],[264,261,259,258,257,256,255,255,255,255,256,258,261,264,269,273,279,284,287,290,292,292,293,292,290,289,287,285,283,280,278,275,273,272,271,270,271,273,275,278,281,284,287,290,292,294,295,295,294,292,290,287,283,280,276,272,269,266,264,263,262,261,261,262,263,265,268,2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    que es lo que se nos pide que demostremos.

    Así pues, LHS=RHS{"x":[[236,236,236,236,236,236,234,233,230,228,227,225,225,224,224,225,238,241,247,251,255,263,271,279,288,296,299,306,315,324,326],[335,336,337,338,339,339,339,338,337,336,335,335,335,335,336,337,338],[344,344,344,344,345,346,347,349,356,361,367,374,379,381,386,391,393,397,400,401,402,403,404,404,404,405,405,404,403,402,400,397,395,394,392,390,390,390,390,390,391,391,392,394,396,399],[511,516,517,518,520,520,519,518,515,511,510,502,499,490,488,480,478,477,474,474,473,473,474,476,477,483,487,491,496,500,504,505,507,508,507,506,500,494,487,483,477,468,464,460,457,452],[611,612,616,618,621,633,637,650,659,663,676,679,688,690,693,696,697],[632,631,634,636,644,649,663,672,681,684,693,696,701,703],[816,816,816,816,816,816,814,811,810,806,804,799,796,794,794,793,792,792,793],[807,807,807,808,812,818,825,829,837,847,850,858,859,860,861,861,860,859,857,853,847,844,834,831,822,817,816,815,815,817,818,823,826,833,839,845,851,857,862,864,871,875,877,879],[936,940,941,941,941,940,937,936,934,933,932,931,930,930,929,929],[935,934,935,937,938,946,948,958,962,972,978,984,990,992,998,1000,1005,1007,1008,1008,1008,1007,1004,1002,1000,998,998,996,995,994,993,991,990,988,987,986,985,983,983,983,983],[1118,1129,1131,1132,1131,1131,1124,1121,1114,1103,1100,1088,1085,1076,1071,1068,1067,1067,1069,1071,1079,1090,1096,1101,1105,1107,1108,1108,1105,1103,1093,1088,1073,1067,1061,1050,1040,1036,1026,1024]],"y":[[242,241,240,239,240,244,251,261,285,294,309,329,336,347,369,382,401,402,404,404,404,404,403,402,400,399,398,397,396,395,394],[263,265,269,279,292,299,317,322,337,344,350,356,359,362,365,365,365],[326,325,324,322,321,319,318,317,313,309,306,302,297,295,291,284,282,276,272,269,268,266,263,260,258,255,254,252,252,253,257,268,277,282,291,309,314,329,338,345,348,351,354,359,361,364],[280,274,271,269,263,261,255,254,250,248,248,249,250,255,258,267,269,272,278,281,285,291,294,297,299,307,312,318,323,328,332,334,339,341,345,346,350,352,352,352,352,352,351,349,348,345],[285,285,285,285,285,285,285,284,284,284,283,283,284,284,285,286,287],[322,322,321,321,320,319,317,315,314,313,312,312,312,312],[248,241,238,241,247,251,261,278,285,307,314,332,342,349,352,358,363,364,365],[254,252,247,244,238,232,228,226,222,219,219,221,222,224,229,236,243,247,250,258,266,270,281,285,293,298,299,302,303,305,307,310,311,315,317,320,324,327,329,330,334,335,337,338],[239,235,238,241,252,257,276,288,298,303,308,311,320,323,331,332],[290,291,291,291,291,290,289,287,286,282,279,276,271,269,262,259,253,249,246,245,242,242,241,243,247,248,251,258,262,265,269,286,292,310,321,331,335,346,348,353,354],[258,245,241,241,238,237,235,235,235,238,240,247,249,258,264,270,276,282,287,289,295,304,308,312,316,320,323,325,329,329,334,335,338,339,339,339,339,339,337,336]],"t":[[0,23,28,33,49,58,66,83,91,103,108,119,124,136,141,158,188,195,202,210,213,219,224,236,241,252,258,266,274,291,293],[544,574,583,591,599,608,618,625,635,641,652,658,668,674,685,691,702],[797,808,818,825,833,841,843,852,858,868,874,885,892,902,908,918,925,935,941,952,958,959,969,975,985,991,1002,1010,1025,1033,1041,1052,1058,1067,1075,1085,1091,1102,1108,1119,1125,1125,1135,1141,1152,1158],[1419,1438,1444,1450,1458,1469,1475,1485,1491,1502,1508,1519,1525,1535,1541,1552,1558,1559,1569,1575,1575,1586,1591,1592,1602,1608,1619,1625,1636,1641,1650,1658,1666,1675,1685,1692,1702,1708,1717,1725,1727,1736,1741,1742,1753,1758],[2188,2207,2212,2213,2217,2225,2236,2242,2253,2258,2269,2275,2286,2292,2292,2303,2309],[2452,2460,2492,2500,2509,2519,2525,2536,2542,2553,2559,2569,2575,2579],[2911,2929,2938,2967,2975,2984,2992,3000,3009,3019,3026,3034,3044,3050,3059,3067,3075,3086,3092],[3248,3253,3260,3267,3276,3286,3292,3300,3309,3320,3326,3336,3342,3343,3353,3359,3367,3376,3378,3386,3392,3403,3409,3420,3426,3436,3442,3453,3459,3470,3476,3487,3492,3501,3509,3519,3526,3536,3542,3553,3559,3570,3576,3586],[3882,3901,3917,3926,3934,3942,3951,3959,3967,3976,3982,3986,3994,4002,4020,4026],[4165,4177,4192,4203,4209,4220,4226,4237,4243,4254,4259,4270,4276,4287,4292,4301,4309,4320,4326,4337,4342,4354,4370,4376,4387,4393,4393,4404,4409,4410,4421,4426,4437,4442,4453,4459,4470,4476,4486,4493,4497],[4869,4889,4896,4901,4909,4920,4926,4937,4943,4954,4959,4971,4976,4987,4993,5004,5009,5021,5031,5034,5043,5059,5068,5076,5086,5093,5104,5109,5121,5126,5137,5143,5154,5159,5160,5171,5176,5188,5193,5201]],"version":"2.0.0"}.

    ¿Es 1+sinx1-sinx=cosx{"x":[[155,151,150,149,149,149,149,149,150,151,152,153,154,155,156],[191,192,194,198,204,211,218,224,229,233,236,239,241,243,244,245,246,245],[219,219,219,218,218,218,218,218,218,218,218,217,217,216,215,214,213,213,213,213,214,215,216],[331,331,332,332,332,332,332,332,331,330,327,324,321,317,312,307,302,298,296,294,294,294,296,298,301,305,308,312,315,319,321,324,325,326,326,326,325,324,322,320,318,315,313,311,309,308,308,308,308,309],[349,348,347,346,345,345,345,345,346,346,346,345,345,345,344,344,344,344,345,345],[353,354,354,354,354,354,354,354,354],[376,376,376,376,376,376,376,376,376,376,376,377,377,377,377,377,377,378,378,378,379,380,381,383,385,387,390,392,395,396,397,398,399,400,402,403,404,405,406,406,406,406,406,406,407,408,410,413,416],[442,442,442,442,442,442,442,442,443,444,446,447,450,452,454,456,458,459,460,460,460,459,458,456,454,453,451,450,449,448,448,450,451,454,456,459,462,464,466,468,470,471,471,470,469,468,466,464,463,462,461,461,462,463,465,468,471,475,479,482,486,489,491],[503,504,505,506,508,510,513,514,516,516,516,515,513,510,507,504,500,497,494,493,493],[132,132,132,131,130,128,126,124,121,117,114,109,105,102,99,98,98,98,100,102,106,109,114,121,127,134,142,146],[593,593,593,593,592,590,589,587,583,581,577,572,567,563,558,555,553,551,551,553,555,557,560,563,566,571,573],[616,616,616,615,614,612,611,610,609,608,608,608,608,609,609,610,610,611],[725,725,725,724,723,721,719,716,712,709,706,704,702,701,700,700,700,701,703,705,708,711,714,717,720,722,724,724,725,724,723,721,719,716,713,710,708,705,704,703],[742,742,741,741,741,740,740,740,740,739,739,739,739,740,740,741,742,743,744,746,749,751],[749,749,749,750],[767,767,768,769,769,769,770,770,770,770,770,769,769,768,768,768,767,767,767,767,767,767,767,768,769,770,772,773,775,777,779,781,783,785,786,788,789,790,790,791,791,791,791,791,791,791,790,790,790,790,791,793,794],[644,643,642,643,645,647,650,653,657,661,665,669,672,675,677,679,680,679,678,676],[820,819,819,818,818,819,820,822,824,827,829,831,832,833,834,834,834,833,832,830,829,828,827,826,825,823,824,825,827,829,831,833,835,837,839,841,842,843,843,843,842,842,841,840,839,839,839,839,841,843,846,850,854,858,861,865,867],[872,873,874,876,879,882,885,887,889,889,888,887,885,883,880,877,875,872,870,867,866,865],[907,906,906,907,908,910,913,916,919,922,926,929,932,935,937,938],[916,915,915,917,919,921,924,928,931,935,939,942,945],[995,994,993,992,991,989,988,987,986,984,983,982,981,981,981,981,982,984,985,988,991,994,997,1000,1002,1005,1007,1008,1009,1010,1011,1011,1011,1011,1010,1009,1008,1007,1006,1006,1005,1005,1005,1005,1006,1008,1010,1013,1016,1019,1022,1024,1026,1028,1029,1029,1029,1029,1029,1029,1028,1026,1025,1024,1022,1021,1019,1018,1017,1016,1015,1015,1014,1014,1015,1016,1018,1021,1025,1028,1033,1037,1042,1046,1049,1052,1054,1056,1057,1059,1060,1061,1061,1061,1061,1061,1060,1059,1057,1056,1054,1053,1052,1052,1053,1054,1056,1058,1060,1063,1065,1067,1069,1070,1071,1071,1070,1068,1066,1063,1060,1057,1056],[1087,1087,1087,1088,1089,1090,1090,1091,1091,1092,1092,1093,1093,1093,1092,1092,1091,1089,1088,1087,1086,1087,1088,1089,1092,1094,1096,1098,1101,1103,1105,1106,1107,1107,1107,1107,1106,1105,1104,1103,1101,1099,1097,1095,1094,1093,1093,1094,1096,1100,1104,1110,1116,1123,1128],[598.8306878306878],[583.5714285714286],[583.5714285714286,583.5714285714286]],"y":[[237,285,302,309,315,321,326,330,334,337,338,340,340,339,338],[293,293,293,293,292,291,290,290,289,289,288,288,288,287,287,287,286,286],[275,273,271,270,269,268,267,269,272,275,280,285,290,296,302,307,312,315,318,319,320,320,319],[273,272,270,269,268,267,265,264,261,260,258,257,257,257,258,260,262,264,266,268,269,271,273,275,276,278,280,282,284,285,287,289,291,293,295,296,298,299,300,300,301,301,301,300,299,298,296,295,294,293],[257,257,258,260,264,269,275,280,286,290,295,299,302,304,306,307,308,309,308,307],[231,228,227,226,225,226,228,230,231],[277,276,275,273,271,270,269,268,269,270,272,275,278,281,284,287,289,292,294,296,297,296,294,291,288,284,281,278,276,274,274,274,274,276,277,279,282,285,288,292,296,300,304,307,309,311,312,312,312],[274,273,272,271,270,269,268,267,266,265,264,264,263,263,263,265,266,269,272,275,278,281,285,288,291,293,295,297,298,299,298,297,295,292,290,287,283,280,277,275,273,271,270,271,272,275,277,281,284,287,291,294,297,299,301,301,302,301,300,299,297,295,294],[246,245,246,248,251,255,261,267,274,281,287,294,301,308,314,320,324,328,331,333,334],[236,235,234,234,233,233,233,234,236,240,245,252,260,269,278,287,295,304,313,321,330,338,346,353,358,363,368,369],[249,248,247,246,246,246,246,246,248,249,252,256,260,266,273,279,286,292,297,302,307,311,315,319,322,325,326],[266,267,269,272,277,283,289,295,300,304,307,310,313,315,317,319,320,320],[283,279,278,276,275,274,273,272,272,272,272,273,274,275,276,277,279,280,282,283,285,287,288,290,292,294,296,298,300,302,304,306,308,310,311,312,313,313,312,311],[276,277,277,278,279,280,282,284,285,289,290,294,298,301,304,305,306,307,307,306,304,303],[246,245,242,242],[286,284,283,282,281,280,280,279,280,281,282,284,286,288,290,292,293,295,297,298,299,300,301,301,301,300,298,296,294,291,289,286,284,282,281,280,279,280,281,282,284,286,289,292,295,298,301,303,305,307,308,309,309],[301,300,300,300,300,300,300,300,300,300,300,300,299,299,299,298,298,299,299,300],[287,287,286,286,285,285,285,284,284,284,284,284,284,285,287,289,292,295,298,301,304,306,307,308,309,309,308,307,306,304,303,301,298,296,293,291,289,288,287,286,287,288,291,294,297,300,303,305,308,309,310,310,310,309,308,307,307],[257,257,259,263,268,273,279,284,290,294,299,305,309,314,318,322,326,329,332,334,335,336],[287,286,285,285,285,284,283,283,282,281,281,280,280,281,281,282],[305,304,303,303,302,302,301,300,299,299,298,298,297],[286,285,284,284,284,284,284,285,287,289,292,294,297,300,303,306,307,309,309,309,309,309,308,307,306,304,303,301,300,298,297,295,294,293,293,293,293,294,295,297,298,300,302,304,306,308,310,311,312,312,312,311,310,308,307,305,303,301,298,296,294,292,290,289,288,288,288,288,288,289,290,292,294,296,297,299,301,302,303,303,303,303,303,302,301,300,299,298,297,296,294,292,290,288,286,284,283,282,282,282,282,282,282,283,284,285,287,289,291,293,296,298,300,302,304,305,307,308,309,310,311,312,312],[293,292,291,291,291,291,292,293,295,296,298,300,302,305,307,309,311,313,314,315,315,315,315,314,312,311,310,308,306,304,302,301,299,298,297,296,296,295,295,295,295,296,298,300,303,307,310,313,317,319,320,320,320,318,317],[525.8888888888889],[523.7089947089947],[523.7089947089947,521.5291005291006]],"t":[[0,13,17,27,35,44,51,60,68,77,84,94,101,109,117],[439,476,484,494,501,512,519,529,534,544,551,561,568,577,584,594,601,626],[768,776,785,794,801,811,818,851,862,868,878,884,894,901,911,918,931,934,946,951,961,968,975],[1344,1346,1352,1355,1363,1363,1372,1378,1385,1395,1401,1411,1419,1429,1437,1450,1451,1463,1468,1478,1485,1495,1501,1511,1519,1529,1538,1549,1551,1563,1568,1578,1585,1595,1601,1611,1620,1629,1638,1649,1653,1663,1668,1678,1685,1695,1701,1711,1718,1725],[1927,1944,1951,1965,1968,1978,1985,1995,2002,2012,2018,2030,2035,2050,2051,2063,2068,2078,2095,2100],[2247,2256,2263,2264,2272,2293,2302,2311,2317],[2535,2538,2548,2552,2560,2568,2578,2585,2610,2618,2627,2635,2644,2652,2663,2669,2679,2686,2695,2702,2719,2735,2745,2752,2763,2769,2779,2785,2795,2802,2812,2819,2829,2835,2846,2852,2863,2869,2878,2885,2895,2902,2912,2919,2929,2936,2947,2953,2963],[3959,3968,3977,3977,3986,3997,4002,4012,4019,4029,4036,4044,4053,4065,4069,4081,4086,4096,4102,4113,4119,4129,4136,4147,4152,4166,4169,4180,4186,4196,4227,4236,4246,4252,4263,4269,4279,4286,4296,4302,4313,4319,4329,4369,4378,4386,4396,4402,4413,4419,4430,4436,4446,4452,4463,4469,4480,4486,4496,4502,4513,4519,4526],[4856,4869,4886,4896,4903,4911,4919,4930,4936,4946,4953,4963,4969,4980,4986,4996,5003,5013,5019,5030,5036],[5721,5735,5753,5762,5778,5786,5797,5803,5813,5820,5830,5836,5847,5853,5864,5870,5880,5886,5897,5903,5913,5920,5930,5936,5947,5953,5964,5969],[6621,6625,6629,6640,6657,6665,6673,6674,6682,6687,6696,6703,6713,6720,6731,6737,6747,6753,6764,6770,6781,6787,6797,6803,6814,6820,6827],[7208,7228,7237,7248,7254,7265,7270,7281,7287,7297,7304,7314,7320,7331,7337,7348,7354,7361],[8399,8411,8418,8421,8431,8437,8448,8454,8465,8471,8481,8487,8498,8504,8515,8521,8532,8537,8548,8554,8565,8571,8582,8587,8598,8604,8615,8621,8632,8637,8648,8654,8665,8671,8682,8687,8698,8704,8715,8720],[8940,8947,8951,8953,8957,8966,8974,8983,8983,8991,9000,9004,9016,9021,9033,9037,9049,9054,9066,9071,9082,9087],[9264,9272,9276,9291],[9635,9642,9646,9648,9654,9665,9671,9688,9704,9715,9721,9732,9738,9749,9754,9766,9771,9782,9788,9799,9804,9816,9821,9832,9838,9849,9854,9866,9871,9882,9888,9899,9904,9916,9921,9932,9938,9954,9965,9971,9982,9988,9999,10005,10016,10021,10032,10038,10049,10055,10066,10071,10078],[13463,13473,13499,13517,13523,13534,13539,13551,13556,13568,13572,13584,13589,13601,13606,13618,13623,13656,13664,13673],[20199,20209,20213,20236,20242,20258,20269,20275,20285,20292,20302,20308,20320,20325,20336,20342,20353,20358,20370,20375,20386,20392,20403,20408,20420,20425,20475,20487,20492,20504,20508,20520,20525,20536,20542,20553,20558,20570,20575,20583,20617,20625,20636,20642,20653,20658,20670,20675,20687,20692,20703,20709,20720,20725,20737,20742,20749],[21148,21184,21192,21202,21209,21220,21225,21237,21242,21253,21259,21270,21275,21287,21292,21304,21309,21320,21325,21337,21342,21353],[43086,43099,43108,43109,43117,43125,43133,43144,43150,43162,43167,43178,43183,43195,43200,43207],[43443,43454,43458,43475,43484,43492,43500,43511,43517,43528,43534,43545,43550],[44014,44021,44025,44027,44034,44045,44050,44062,44067,44076,44084,44095,44101,44112,44117,44129,44134,44145,44151,44162,44167,44179,44184,44196,44201,44212,44217,44229,44234,44245,44251,44262,44267,44279,44284,44301,44311,44317,44329,44334,44346,44351,44362,44367,44379,44384,44396,44401,44412,44417,44429,44434,44445,44451,44462,44467,44479,44484,44495,44501,44512,44517,44529,44534,44546,44551,44562,44567,44579,44584,44596,44601,44612,44617,44629,44634,44646,44651,44662,44667,44679,44684,44693,44701,44711,44717,44729,44734,44746,44751,44762,44767,44779,44784,44796,44803,44812,44817,44829,44834,44846,44851,44862,44884,44892,44901,44911,44917,44929,44934,44946,44951,44962,44967,44979,44984,44995,45003,45011,45018,45029,45034,45041],[45401,45405,45410,45418,45434,45451,45462,45468,45480,45484,45496,45501,45513,45518,45530,45534,45546,45551,45563,45568,45584,45601,45612,45618,45630,45634,45647,45651,45663,45668,45680,45684,45696,45701,45713,45718,45730,45734,45746,45751,45763,45768,45780,45784,45797,45801,45809,45818,45829,45834,45846,45851,45863,45868,45875],[1646821971835],[1646821972403],[1646821972549,1646821972680]],"version":"2.0.0"} una identidad trigonométrica válida?

    Solución:

    Paso 1: El LHS parece más complicado que el RHS, así que simplifiquemos el LHS y comprobemos si llegamos al RHS o no.

    Paso2: Se puede ver que el LHS se puede simplificar utilizando la identidad algebraica.

    a+ba-b=a2-b2{"x":[[159,158,157,157,157,157,158,158,158,159,159,157,156,153,150,146,143,139,135,132,129,127,126,126,126,127,129,130,133,136,139,143,147,150,154,158,161,163,165,167,168,169,170,170,171,172,172,173,174,175,177,178,181],[206,208,211,216,223,230,237,243,248,253,256,258,260,261],[233,233,232,233,234,235,236,236,236,236,236,236,235,234,233,233,233,234,235],[291,290,289,289,288,287,286,285,284,284,284,284,284,284,284,284,284,284,285,285,285,286,288,290,292,294,296,299,301,304,307,310,312,313,314,314,312,310,306,301,297,292,287,283,279,276,273,271,269,269],[348,349,350,352,354,357,359,360,360,358,355,352,348,342,340],[119,119,119,119,119,119,118,117,116,113,110,105,99,93,89,86,85,87,91,97,101],[432,430,429,426,425,421,418,413,408,403,398,392,388,385,383,384,385,388,392],[449,449,449,449,449,449,449,449,447,446,444,441,437,434,431,428,426,424,424,424,425,427,429,431,434,437,440,444,447,450,453,455,456,457,458,460,461,462,463,464,465,466],[499,498,497,496,497,499,502,507,512,517,522,527,531],[566,566,566,566,566,565,564,563,561,560,559,558,558,558,558,558,558,558,559,559,559,560,561,562,564,565,566,568,570,571,573,574,576,577,578,579,579,579,578,576,574,570,566,562,559,557,556],[597,597,597,598,599,603,607,610,614,616,616,615,613,609,605,601,598,595,593],[665,665,664,664,665,668,672,676,681,685,690,693,695,696],[657,656,655,654,655,658,662,667,673,680,687,690],[761,761,761,762,762,763,764,764,764,763,761,759,756,753,749,746,742,740,738,737,737,739,741,743,746,750,753,757,761,764,766,768,769,769,769,769,768,768,768,769,771,773,776,779,783,788],[806,805,805,804,804,804,803,803,804,806,809,812,817,821,824,826,827,828,828,826,825,823,821,820,820,820,822,825,829,834,839,844,847],[850,849,846,845,847,850,854,860,867,874,882,889,894,897,899],[947,948,948,949,949,949,949,947,946,945,943,941,940,939,938,938,937,937,937,936,936,936,936,936,937,938,940,942,944,947,950,954,957,959,960,960,958,956,951,946,942,937,934,933,932],[972,972,972,973,974,976,980,981,983,988,992,996,997,999,999,999,997,994,991,988,985,982,980,980,981,985,991,1000,1005]],"y":[[292,292,292,291,290,289,286,284,280,277,273,270,268,267,267,269,272,276,282,288,295,301,307,311,313,314,314,313,311,308,304,300,295,290,285,281,278,277,276,277,279,281,284,287,291,293,296,299,300,302,302,302,302],[286,285,285,285,284,283,281,280,278,277,277,276,276,276],[271,269,268,268,270,273,276,280,284,290,294,299,303,306,309,311,312,313,313],[230,230,231,232,236,242,249,257,263,269,274,280,284,289,293,295,297,298,298,297,296,293,289,285,281,278,276,274,272,272,273,274,277,279,283,286,291,295,299,303,306,308,309,309,309,307,304,300,295,293],[231,231,232,236,238,246,255,264,273,282,293,302,312,322,326],[254,253,252,251,250,249,248,247,247,248,251,257,266,277,290,303,313,322,331,340,344],[225,224,224,223,223,223,226,231,238,247,258,270,281,292,302,310,317,324,329],[279,278,277,275,273,271,270,268,266,265,265,265,266,269,272,277,281,285,289,291,292,292,292,292,291,289,286,284,281,279,276,275,274,274,274,275,277,280,283,286,289,293],[284,284,283,283,283,283,283,282,282,282,282,281,281],[244,241,236,235,234,234,234,235,238,242,247,252,257,263,268,273,278,282,285,287,288,288,288,286,284,282,280,278,276,275,274,273,274,275,278,282,286,289,293,297,299,301,302,302,301,299,297],[234,233,231,232,233,236,241,246,254,262,270,279,288,297,305,314,320,326,329],[268,267,266,265,265,265,265,265,266,266,266,267,268,268],[292,292,291,291,291,292,293,293,294,294,294,293],[286,285,284,283,282,279,277,274,271,269,267,267,267,268,270,274,278,282,286,290,293,295,296,296,295,294,292,289,286,283,280,278,277,276,277,279,282,286,290,295,298,302,304,305,305,305],[223,220,217,216,215,214,213,212,211,210,209,208,207,206,207,208,211,214,217,221,225,228,231,233,235,236,236,236,236,235,234,234,233],[277,277,276,276,276,276,276,276,275,275,274,274,273,273,274],[244,243,242,240,239,240,243,246,250,255,260,266,271,276,281,284,287,289,290,291,290,289,287,285,284,282,280,279,278,278,278,280,283,287,290,294,297,299,302,303,304,305,305,305,304],[217,214,209,208,208,206,206,205,205,204,204,204,204,206,208,211,214,218,221,224,227,229,231,232,233,233,233,233,232]],"t":[[0,5,15,56,65,75,82,91,98,108,115,125,132,141,148,158,165,175,182,192,198,211,215,227,232,241,248,258,265,275,282,293,298,308,315,325,332,344,348,361,365,373,382,391,398,409,415,425,432,441,448,458,465],[784,807,815,825,832,842,849,858,865,875,882,892,899,915],[1093,1105,1110,1113,1124,1132,1142,1149,1160,1165,1176,1182,1192,1199,1209,1215,1226,1232,1242],[1481,1489,1498,1499,1509,1516,1525,1532,1542,1549,1559,1566,1576,1582,1592,1599,1611,1616,1632,1642,1649,1659,1666,1676,1682,1692,1699,1709,1716,1726,1732,1742,1749,1759,1766,1776,1782,1792,1799,1809,1816,1826,1832,1843,1849,1859,1866,1876,1882,1890],[2185,2196,2201,2204,2209,2216,2226,2233,2243,2249,2259,2266,2276,2282,2290],[2859,2866,2871,2878,2886,2894,2903,2911,2916,2927,2933,2943,2949,2960,2967,2976,2983,2993,2999,3010,3015],[3542,3549,3554,3556,3561,3569,3577,3583,3591,3600,3610,3616,3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    Aplicando esta identidad respectivamente, obtenemos

    1+sinx1-sinx=1-sin2x{"x":[[70,70,69,68,66,65,63,58,56,54,50,45,42,40,39,40,42,47,53,61,68],[95,95,94,93,93,92,91,91,91,91,91,92,93,93,94,94],[113,112,112,111,114,118,123,132,135,138,141],[129,128,128,129,129,130,130,130,131,131,131,131,132,132],[177,177,177,177,177,176,175,174,170,166,162,161,159,159,160,163,167,172,174,177,180,180,180,177,176,174,171,170,166,164,163,163],[193,192,191,191,192,192,192,192,192,192,192,192,193,193],[199,197,197,199],[208,207,207,207,207,208,208,209,209,209,209,209,210,210,210,210,211,211,213,216,218,222,223,225,226,226,227,227,228,229,230,231],[247,246,245,245,244,244,245,246,249,252,253,255,258,258,258,258,258,257,255,253,253,252,251,251,252,254,255,257,260,262,263,264,264,265,265,266,266,267,269,269,270,272],[286,284,283,284,285,288,289,294,299,300,301,301,301,299,297,296,292,289,287,286],[337,336,336,334,331,330,329,327,326,323,319,316,315,313,312,313,316,320,325,331],[351,351,349,349,349,349,348,347,347,347,347,347,347,348,348,349],[371,371,370,371,374,377,382,387,393,398,403,405,403],[439,439,439,439,439,439,439,439,437,435,433,430,427,425,424,424,425,426,429,432,435,437,438,438,437,435,432,430,428,426,425,425,425],[451,451,451,451,452,452,452,451,451,450,450,450,449,449,449,449,450],[451,451,451,451,452,453,454],[465,464,464,464,465,465,466,466,466,466,466,466,466,466,466,467,467,468,470,471,472,475,477,479,480,481,482,483,483,483,483,484,484,484,485,485,486,487],[508,507,507,506,505,505,507,509,512,515,519,522,524,525,525,524,522,520,518,515,514,513,512,513,515,517,520,524,527,530,533,534,534,533,532,531,531,530,530,531,532,533,535,537,540],[555,555,555,555,557,559,562,566,569,572,573,573,571,568,565,563,561,560,559,559,560,561],[607,606,606,607,608,611,614,617,621,624,627,628],[603,602,601,602,606,610,616,622,628,632,636,639,640],[719,719,718,718,718,718,718,718,718,718,718,717,717,718,719,720],[740,739,739,740,742,746,752,759,765,769,772,774,773,772],[833,834,835,835,836,836,836,835,833,831,828,826,824,822,821,820,820,822,825,828,831,834,835,835,835,832,830,826,823,821,818,816,815,815,815],[848,848,849,849,849,849,848,848,847,847,847,847,847,849,851],[858,857,857,855,854,855,856,857],[865,865,864,864,864,864,865,865,865,865,865,865,865,865,866,866,867,868,870,872,875,877,880,882,884,886,887,887,887,887,886,885,885,885,885,885,887,888],[902,901,898,897,897,896,896,898,900,903,907,910,913,914,914,914,911,908,905,902,900,897,896,896,898,902,908,914,919,922],[937,935,933,933,932,932,932,932,933,935,938,942,945,949,951,952,952,951,948,945,943,940,939,938,939,941,944,947,951,954,957,959,960,960,959,957,954,952,950,948,947,947,949,952,957,960]],"y":[[224,222,221,219,217,217,217,219,220,222,228,236,245,254,263,271,277,280,282,282,281],[231,229,228,227,229,233,239,241,246,251,256,260,266,267,268,269],[248,248,247,247,247,247,246,244,243,242,242],[237,236,235,235,236,236,238,239,243,246,250,254,257,259],[238,237,232,230,229,228,227,226,226,228,231,233,235,238,240,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,256,255],[237,235,234,236,239,240,242,243,246,248,250,253,256,258],[220,218,217,219],[241,239,238,237,236,236,237,238,241,242,244,248,250,252,253,251,250,248,245,240,237,234,233,233,235,238,239,241,243,248,252,255],[240,239,239,238,238,237,236,236,236,236,236,236,238,239,242,246,247,249,252,253,254,254,254,252,251,248,246,244,241,240,240,241,242,243,245,247,248,251,252,253,253,253],[213,212,210,211,213,215,217,222,229,233,236,240,247,255,258,261,268,275,280,283],[219,218,217,215,214,214,214,215,216,221,227,236,240,249,258,264,268,272,273,274],[234,233,231,230,232,233,239,244,246,249,257,261,264,267,268,268],[247,246,245,245,245,245,244,244,243,242,241,240,240],[239,238,237,235,234,232,230,229,227,227,227,229,231,234,236,239,241,243,245,246,247,248,248,249,250,250,251,251,251,251,250,249,248],[236,234,233,232,231,230,231,234,237,240,243,246,249,252,255,258,259],[222,221,220,219,220,221,222],[241,239,238,237,237,238,240,242,244,247,249,251,252,253,254,254,252,250,247,244,242,239,237,236,236,237,239,240,241,243,246,249,251,253,255,256,256,256],[239,238,237,236,235,234,233,232,232,232,231,232,233,235,238,240,243,246,249,251,252,253,253,251,250,247,245,242,240,238,236,235,234,235,237,238,241,243,245,247,249,250,251,251,251],[212,211,210,209,209,209,210,212,216,220,224,230,235,240,245,250,254,257,260,262,263,263],[226,225,224,224,223,223,223,223,224,225,227,227],[242,242,242,242,242,241,239,238,238,237,237,237,237],[216,215,212,213,215,219,223,228,233,235,240,244,248,250,252,252],[228,227,226,225,225,224,224,223,222,222,222,222,222,221],[223,222,220,219,217,215,213,211,210,210,210,211,212,214,217,220,223,225,228,230,232,233,234,236,237,237,238,238,238,238,238,237,236,234,233],[221,220,218,217,219,222,224,227,230,232,235,237,238,239,239],[205,204,203,202,201,203,204,205],[226,225,224,223,224,226,228,229,232,233,235,237,238,239,239,238,237,234,231,228,224,221,219,218,218,218,219,221,224,227,231,234,237,240,243,245,246,246],[187,186,182,180,179,178,176,175,173,172,171,171,171,173,175,178,182,185,188,191,194,196,197,198,198,198,197,196,195,195],[222,221,219,218,217,216,215,214,213,213,212,212,213,214,216,219,222,225,227,230,232,233,234,234,234,233,231,229,226,224,222,220,219,218,218,219,220,222,224,227,229,231,233,235,235,236]],"t":[[0,5,12,20,31,37,48,53,64,65,71,80,87,97,103,114,120,130,137,147,153],[577,584,589,597,615,620,630,637,647,655,663,670,680,688,688,698],[805,812,815,820,845,854,863,870,880,888,889],[1027,1034,1039,1064,1070,1071,1080,1089,1097,1104,1114,1123,1130,1135],[1364,1376,1380,1382,1387,1397,1404,1414,1421,1431,1437,1447,1454,1464,1471,1481,1487,1498,1506,1514,1521,1531,1543,1547,1555,1566,1567,1575,1581,1587,1597,1605],[1751,1756,1771,1798,1804,1814,1824,1825,1831,1837,1838,1850,1854,1864],[1996,2003,2006,2029],[2462,2479,2483,2488,2498,2521,2532,2538,2548,2554,2564,2571,2581,2598,2604,2629,2632,2639,2648,2654,2665,2681,2688,2700,2704,2715,2724,2725,2731,2738,2754,2764],[2962,2968,2972,2974,2978,2985,3008,3009,3020,3022,3032,3038,3048,3056,3065,3071,3081,3090,3098,3104,3115,3116,3132,3148,3154,3165,3173,3182,3188,3204,3215,3221,3233,3234,3238,3248,3255,3265,3271,3282,3288,3298],[3459,3469,3473,3491,3498,3505,3515,3521,3532,3539,3549,3550,3555,3565,3574,3575,3582,3588,3598,3609],[4173,4180,4189,4199,4215,4222,4223,4234,4235,4244,4249,4255,4265,4273,4282,4288,4299,4305,4315,4322],[4671,4683,4690,4690,4710,4713,4722,4732,4740,4749,4755,4766,4772,4782,4788,4799],[4954,4965,4969,4990,4999,5005,5016,5022,5032,5039,5049,5055,5088],[5456,5465,5468,5472,5483,5489,5499,5505,5516,5522,5532,5539,5549,5555,5566,5572,5582,5591,5599,5605,5616,5622,5633,5639,5649,5655,5666,5672,5683,5689,5699,5707,5721],[5845,5851,5855,5857,5864,5872,5897,5905,5914,5922,5933,5939,5950,5956,5966,5972,5983],[6121,6128,6132,6134,6156,6164,6172],[6320,6327,6332,6334,6364,6372,6383,6391,6400,6406,6417,6422,6433,6439,6450,6456,6466,6472,6483,6489,6500,6506,6516,6522,6533,6539,6550,6556,6557,6567,6572,6583,6589,6600,6606,6617,6622,6633],[6844,6849,6853,6856,6867,6883,6889,6900,6906,6917,6923,6934,6939,6951,6956,6967,6973,6984,6989,7001,7006,7017,7023,7039,7050,7056,7067,7073,7084,7089,7100,7106,7117,7140,7150,7156,7168,7173,7184,7189,7201,7206,7218,7223,7234],[7479,7490,7494,7499,7506,7517,7523,7534,7540,7551,7556,7568,7573,7584,7589,7601,7606,7615,7623,7633,7640,7647],[8098,8111,8123,8140,8150,8156,8167,8173,8184,8190,8201,8205],[8333,8338,8344,8373,8383,8390,8401,8406,8415,8423,8434,8440,8447],[8879,8890,8895,8912,8920,8928,8937,8945,8953,8957,8968,8973,8984,8990,9001,9007],[9202,9213,9218,9221,9226,9234,9240,9251,9257,9268,9274,9284,9317,9323],[9811,9821,9825,9827,9832,9840,9852,9857,9868,9874,9883,9890,9901,9907,9918,9924,9934,9940,9952,9957,9965,9974,9984,9990,10002,10007,10019,10024,10036,10040,10053,10057,10068,10074,10085],[10255,10265,10269,10279,10304,10316,10324,10335,10341,10352,10357,10369,10374,10385,10391],[10496,10502,10507,10509,10516,10549,10557,10565],[10738,10741,10745,10750,10782,10791,10802,10807,10820,10824,10836,10841,10852,10857,10869,10874,10886,10891,10902,10907,10919,10924,10936,10941,10952,10957,10969,10974,10986,10991,11002,11007,11016,11024,11035,11041,11052,11057],[11317,11332,11337,11339,11346,11352,11358,11369,11374,11387,11391,11404,11408,11420,11424,11436,11441,11453,11458,11469,11474,11486,11491,11503,11508,11519,11524,11536,11541,11548],[11879,11890,11895,11897,11905,11908,11919,11925,11936,11941,11953,11958,11970,11974,11986,11991,12003,12008,12019,12026,12036,12041,12053,12058,12074,12085,12091,12103,12108,12120,12124,12136,12141,12153,12158,12170,12174,12186,12191,12203,12208,12220,12225,12236,12241,12248]],"version":"2.0.0"}

    Paso3: Aplicando ahora la identidad pitagórica para el seno y el coseno, obtenemos

    1-sin2x=cos2x{"x":[[137,137,136,136,136,136,136,135,133,133,132,131,131,131,131,131,132,132,133,133],[169,168,166,167,168,172,181,189,193,201,208,216,217,219,218],[273,273,274,275,275,276,276,276,276,274,272,269,265,261,258,257,257,257,260,264,269,273,276,278,277,274,270,266,263,260,258,257,257],[286,286,285,285,286,287,288,288,289,289,289,289,289,289,289,291,293,297],[301,300,299,298,298,299,300],[309,308,308,308,307,307,307,307,307,307,307,308,308,309,309,310,311,312,313,315,317,319,321,322,323,324,325,325,326,326,326,327,327,327,328,329,331,332,333],[382,382,382,382,383,384,385,387,389,392,394,395,395,395,394,392,390,387,385,384,383,384,387,390,394,399,403,406,408,409,408,408,407,406,405,405,405,405,406,408,411,415],[346,345,344,343,343,342,342,342,342,343,345,347,350,353,356,357,358,358,357,355,352,349,347,345,344,344,346,349,353,357,359],[462,461,459,459,460,463,468,473,479,485,491,495,498,498],[468,467,466,465,466,469,472,476,480,485,489,494,497],[570,570,570,569,568,566,565,563,560,558,555,552,550,549,549,550,552,554,558,562,566,570,574,578,581,583,584,585,586,586,586,586,586,585,585,585,585,585,587,588,590,592,595,597,599,601,602,603,604,604,602,601,599,596,594,592,591,590,590,591,593,597,601,605,611,616,621,625,628,631,632,633,633,632,629,627,624,622,621,620,620,621,624,627,631,635,637,638,638,636,633,630,627,624],[650,649,649,649,649,651,652,654,657,659,662,664,665,665,665,665,664,662,659,657,654,653,653,654,657,662,668,674,680],[682,681,680,680,680,682,684,687,689,692,695,696,696,696,695,692,690,687,685,683,683,683,686,689,693,698,702,705,706,706,705,704,703,701,700,699,699,699,700,702,706,711]],"y":[[206,203,200,198,200,201,204,211,223,227,237,250,254,261,266,268,271,273,274,275],[234,233,231,231,231,231,231,230,229,228,227,226,226,226,226],[223,222,220,219,218,217,215,214,213,213,213,213,215,217,220,222,225,227,229,230,231,232,233,234,235,237,238,240,241,242,242,242,241],[220,219,217,216,216,216,217,219,222,225,229,233,236,239,242,243,243,241],[201,200,200,199,198,199,200],[223,222,221,220,220,222,224,225,228,231,233,236,238,240,241,241,240,237,234,231,227,225,223,221,221,221,223,225,226,228,229,231,232,234,235,238,240,241,241],[214,213,210,209,208,207,206,206,205,206,207,209,212,215,219,222,225,228,230,231,232,230,228,225,222,218,215,212,210,210,210,212,214,217,220,223,226,229,231,233,234,234],[180,178,177,176,175,174,172,171,169,168,167,166,166,166,167,170,173,177,181,184,187,189,191,192,192,191,191,189,188,187,186],[210,209,208,207,207,207,206,205,205,205,205,205,206,207],[223,223,223,223,222,222,221,220,219,219,218,218,218],[211,208,205,205,203,203,202,202,204,205,208,212,217,220,224,227,229,231,232,232,232,231,230,227,225,223,220,218,216,214,213,214,216,219,222,225,228,231,233,234,235,235,234,232,229,226,223,219,216,213,211,209,209,210,211,213,215,218,221,224,226,228,229,229,229,228,225,223,220,217,214,211,209,207,207,206,206,207,208,210,212,215,218,221,225,229,232,235,237,238,239,239,239,238],[176,175,172,170,169,168,166,165,163,163,163,163,164,167,170,173,177,181,184,187,190,191,192,192,192,191,190,189,188],[214,211,209,208,207,206,205,205,205,206,208,210,213,216,219,222,225,227,229,230,231,230,228,226,223,219,216,213,210,209,209,210,212,214,217,220,222,225,228,230,232,233]],"t":[[0,1,6,8,33,33,42,52,58,68,74,85,92,101,108,108,118,125,126,135],[408,414,417,441,443,458,467,475,485,491,501,508,518,525,541],[937,943,947,951,958,971,975,985,991,1001,1008,1018,1025,1035,1041,1051,1058,1068,1075,1085,1092,1101,1108,1118,1134,1142,1151,1158,1168,1175,1185,1191,1201],[1351,1358,1362,1366,1392,1401,1408,1418,1425,1435,1442,1452,1458,1468,1475,1484,1492,1502],[1615,1620,1624,1625,1633,1658,1668],[1810,1817,1821,1823,1834,1863,1871,1875,1885,1892,1902,1908,1919,1925,1935,1942,1952,1958,1969,1975,1985,1992,2002,2009,2019,2025,2034,2042,2046,2054,2055,2063,2064,2071,2072,2080,2088,2096,2102],[2954,2962,2969,2971,2976,2986,2992,3003,3009,3019,3026,3036,3042,3053,3060,3070,3076,3086,3092,3103,3109,3134,3142,3153,3159,3169,3176,3186,3192,3203,3219,3226,3236,3244,3253,3259,3270,3276,3286,3292,3303,3309],[4696,4711,4716,4720,4726,4734,4743,4753,4759,4770,4776,4786,4793,4803,4810,4820,4826,4837,4843,4854,4860,4868,4876,4887,4893,4904,4910,4921,4926,4937,4942],[5435,5442,5447,5449,5470,5476,5485,5493,5504,5510,5521,5527,5538,5542],[5639,5645,5650,5668,5685,5693,5704,5710,5721,5727,5738,5743,5754],[6222,6239,6244,6249,6257,6267,6274,6274,6282,6288,6293,6304,6310,6321,6327,6337,6343,6354,6360,6371,6377,6387,6393,6405,6410,6421,6427,6437,6444,6454,6460,6493,6502,6510,6521,6527,6538,6544,6553,6560,6570,6577,6588,6594,6604,6610,6622,6627,6638,6643,6654,6660,6671,6677,6688,6694,6704,6710,6721,6727,6738,6744,6754,6760,6771,6777,6788,6794,6804,6810,6821,6827,6838,6843,6854,6860,6871,6877,6888,6894,6904,6910,6921,6927,6938,6944,6954,6960,6971,6977,6988,6994,7004,7010],[7250,7256,7260,7262,7269,7277,7288,7294,7305,7310,7321,7327,7338,7344,7355,7361,7371,7377,7388,7394,7405,7411,7422,7427,7438,7444,7455,7460,7470],[7723,7734,7739,7746,7754,7761,7771,7777,7788,7794,7805,7811,7822,7827,7839,7844,7856,7861,7872,7878,7889,7905,7911,7922,7927,7939,7944,7955,7961,7972,7988,7994,8006,8011,8022,8027,8039,8044,8052,8061,8072,8077]],"version":"2.0.0"}

    que no puede simplificarse más.

    Se observa que LHSRHS{"x":[[212,213,214,215,215,215,214,211,210,210,207,206,203,200,198,198,197,197,213,227,233,250,256,273,284,294,300,309,320,323,328],[330,330,330,331,331,331,331,328,327,324,320,318,317,316,316,314,314,314,314],[320,322,324,327,330,343,347,359,363,373,376,380,386,388,391,394,401,403,408,410,411,412,413,413,413,411,410,407,405,403,401,400,397,396,394,393,389,388,385,384,384,384,385,387,388,391,393,396],[531,536,537,538,538,538,534,533,530,527,517,513,501,497,484,476,470,468,466,467,468,476,478,481,488,492,496,503,510,515,517,519,520,523,523,519,517,512,502,497,484,480],[626,650,656,671,677,698,704,714,719,724,728,738,741,744,746],[636,635,634,636,642,646,658,664,680,685,699,703,707,714,718,721,726,728,730,733],[703,703,703,703,703,703,703,703,703,702,701,700,700,697,697,694,693,693,692,691,690,690,690],[831,831,831,831,832,833,835,836,837,838,838,838,838,836,836,833,833,833,833,833,833,833,834,835],[841,840,838,837,836,836,838,841,843,852,856,867,871,883,884,886,888,888,887,886,883,875,868,860,857,848,844,844,844,846,851,854,865,869,873,881,886,888,894,900,902,906,907,908,908,906],[983,981,981,983,984,986,986,988,989,989,989,989,989,988,988,987,987,987,987,987,987,987],[990,988,988,988,990,991,998,1004,1008,1020,1025,1036,1043,1050,1053,1061,1064,1067,1071,1072,1073,1073,1073,1073,1073,1071,1071,1070,1068,1067,1067,1065,1064,1063,1062,1061,1060,1060,1058,1058,1058,1057,1058,1059,1061,1062,1065,1066,1068,1069],[1196,1199,1199,1197,1196,1193,1184,1180,1176,1168,1164,1160,1153,1150,1143,1134,1131,1126,1125,1125,1125,1127,1128,1136,1138,1146,1150,1154,1162,1166,1170,1180,1184,1188,1190,1191,1191,1190,1188,1184,1180,1173,1166,1158,1154,1146,1137,1135,1133]],"y":[[251,244,243,245,248,260,271,284,292,301,317,326,345,369,384,398,410,415,441,444,445,447,447,447,445,445,444,444,443,443,442],[274,273,272,272,277,287,302,321,331,349,371,384,394,399,403,411,417,418,422],[352,352,351,351,349,346,345,341,339,334,332,331,327,324,322,319,312,309,300,294,288,285,277,275,271,265,265,262,262,263,265,267,272,277,281,286,304,311,338,354,368,375,387,400,404,414,417,422],[296,285,281,277,274,272,267,265,264,263,263,264,269,272,283,292,299,304,311,320,322,330,332,335,340,343,345,350,355,361,364,366,368,375,377,383,385,387,388,388,387,386],[312,311,311,312,312,312,312,311,311,310,310,309,309,309,309],[362,362,362,362,362,362,361,361,359,358,357,357,357,357,357,357,356,356,356,356],[274,272,269,265,261,260,261,267,270,275,285,298,305,328,337,359,370,380,385,397,404,406,409],[286,285,284,280,277,276,274,274,278,281,295,308,317,344,354,378,384,398,405,411,412,414,415,415],[294,292,289,287,281,278,275,271,270,265,264,261,259,259,260,261,264,268,272,274,280,291,298,307,311,322,329,334,337,339,345,347,353,355,357,362,365,367,372,378,381,389,391,398,399,404],[269,268,270,276,280,285,291,311,317,338,343,354,359,371,376,390,394,405,408,415,416,417],[355,354,353,352,351,349,347,345,344,340,339,335,331,328,326,319,317,312,304,301,293,291,288,282,277,272,269,268,267,267,268,275,284,294,299,309,327,334,351,357,362,371,379,385,389,391,394,395,395,395],[280,267,265,261,260,259,259,261,262,267,269,271,278,281,289,304,309,322,331,337,339,342,344,349,351,354,355,357,359,360,361,367,371,375,378,382,390,392,395,400,404,408,412,415,416,417,415,414,411]],"t":[[0,17,25,40,49,57,65,77,82,82,93,99,110,115,126,132,143,149,178,193,199,207,216,226,232,243,249,257,265,276,285],[555,566,576,582,593,599,609,616,626,632,643,649,657,666,668,676,682,693,699],[875,899,907,910,916,924,932,943,949,959,966,966,974,982,985,993,999,1010,1016,1026,1032,1043,1049,1059,1066,1076,1083,1093,1099,1110,1116,1116,1126,1133,1133,1143,1149,1160,1165,1176,1182,1193,1199,1210,1216,1226,1232,1243],[1522,1540,1547,1558,1559,1566,1576,1583,1583,1591,1599,1610,1616,1627,1632,1643,1649,1660,1666,1677,1683,1693,1699,1700,1710,1716,1717,1727,1737,1741,1749,1752,1760,1766,1774,1782,1793,1799,1810,1816,1827,1833],[2239,2258,2264,2274,2283,2301,2302,2308,2316,2316,2325,2333,2341,2350,2352],[2509,2525,2533,2550,2560,2566,2577,2583,2594,2600,2610,2616,2617,2625,2633,2635,2644,2649,2650,2661],[2826,2833,2837,2843,2850,2866,2883,2893,2900,2900,2911,2916,2927,2935,2944,2950,2960,2966,2977,2983,2994,3000,3004],[3360,3367,3372,3375,3383,3394,3400,3411,3416,3427,3433,3444,3450,3461,3466,3477,3483,3494,3500,3511,3516,3517,3527,3533],[3689,3694,3701,3708,3717,3727,3733,3744,3750,3761,3767,3777,3783,3794,3800,3800,3811,3816,3825,3834,3842,3850,3858,3867,3877,3883,3894,3900,3911,3917,3927,3933,3944,3950,3950,3961,3967,3967,3978,3983,3994,4000,4011,4017,4028,4033],[10365,10378,10402,10413,10419,10420,10430,10436,10447,10452,10461,10469,10472,10480,10486,10497,10502,10513,10519,10530,10536,10536],[10787,10806,10814,10819,10828,10836,10844,10853,10861,10869,10880,10886,10897,10903,10913,10919,10930,10936,10947,10953,10963,10969,10971,10981,10986,10995,11003,11013,11019,11036,11046,11053,11064,11069,11080,11086,11097,11103,11114,11119,11120,11131,11136,11147,11153,11164,11169,11170,11181,11185],[11575,115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para cualquier valor de x. Pero para que una identidad sea válida, debe satisfacer cada valor para el que está definida la función.

    Por tanto, la identidad trigonométrica dada es falsa.

    Verificación de identidades trigonométricas - Puntos clave

    • Una identidad trigonométrica es verdadera si y sólo si satisface todos los valores para los que está definido el dominio.
    • Si una identidad sólo es cierta para determinados valores, entonces esos valores se llaman soluciones de esa ecuación.
    • Hay dos formas de demostrar una identidad trigonométrica: Algebraicamente y gráficamente.
    • Para demostrar una identidad algebraicamente, simplifica uno de los lados de la identidad reduciéndolo a términos más sencillos con ayuda de las identidades fundamentales.
    • Para el método gráfico, traza las gráficas de ambos lados de la ecuación y si ambas son exactamente iguales, entonces la identidad es cierta.
    • Siempre es más conveniente demostrar una identidad utilizando el método algebraico, ya que es más eficaz y fácil.
    Preguntas frecuentes sobre Verificación de identidades trigonométricas
    ¿Qué son las identidades trigonométricas?
    Las identidades trigonométricas son ecuaciones verdaderas para ángulos específicos, relacionando funciones trigonométricas como seno, coseno y tangente.
    ¿Para qué se utilizan las identidades trigonométricas?
    Se utilizan para simplificar expresiones trigonométricas y resolver ecuaciones en trigonometría, facilitando el cálculo de ángulos y longitudes.
    ¿Cuáles son las identidades trigonométricas básicas?
    Las básicas incluyen las identidades de Pitágoras: sen²(θ) + cos²(θ) = 1, tan(θ) = sen(θ)/cos(θ), y 1 + tan²(θ) = sec²(θ).
    ¿Cómo se verifica una identidad trigonométrica?
    Para verificar una identidad trigonométrica, transforma una parte de la ecuación para que coincida con la otra, usando identidades conocidas y simplificación.

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