Question

(sec theta -cos theta)²-(cosec thetha-sin thetha)²-(cot thetha - tan thetha)²​

Answer 1

★ SoluTion :

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$\sf : \implies \: (sec \theta - cos \theta) {}^{2} - (cosec \theta - sin \theta) {}^{2} - (cot \theta - tan \theta) {}^{2}$

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$\sf : \implies \: (sec {}^{2} \theta + cos {}^{2} \theta - 2sec \theta \: cos \theta) - (cosec \theta + sin \theta - 2 cosec \theta \: sin \theta) - (cot \theta + tan \theta - 2cot \theta \: tan \theta)$

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$\sf : \implies \: \left( sec {}^{2} \theta + cos {}^{2} \theta - 2sec \theta \dfrac{1}{sec \theta}\right) - \left(cosec {}^{2} \theta + sin {}^{2} \theta - 2cosec \theta \dfrac{1}{cosec \theta} \right) - \left(cot {}^{2} \theta \: + tan {}^{2} \theta - 2cot \theta \dfrac{1}{cot \theta} \right)$

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$\sf : \implies \: (sec {}^{2} \theta + cos {}^{2} \theta - 2) - (cosec {}^{2} \theta + sin {}^{2} \theta - 2) - (cot {}^{2} \theta + tan {}^{2} \theta - 2)$

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$\sf : \implies \: sec {}^{2} \theta + cos {}^{2} \theta - 2 - cosec {}^{2} \theta - sin {}^{2} \theta + 2 - cot {}^{2} \theta - tan {}^{2} \theta + 2$

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$\sf : \implies \: sec {}^{2} \theta \: + cos {}^{2} \theta - cosec {}^{2} \theta - sin {}^{2} \theta - cot {}^{2} \theta - tan {}^{2} \theta + 2$