Question

____¿¿¿¿____
Prove that

$\sec( \frac{3\pi}{2} - \alpha ) \sec( \alpha - \frac{5\pi}{2} ) + \tan( \frac{5\pi}{2} + \alpha ) \tan( \alpha - \frac{3\pi}{2} ) = - 1$


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Answer 1

Question:-

Prove that,

$\sf\sec( \frac{3\pi}{2} - \alpha ) \sec( \alpha - \frac{5\pi}{2} ) + \tan( \frac{5\pi}{2} + \alpha ) \tan( \alpha - \frac{3\pi}{2} ) = - 1$

Proof:-

$\sf\sec( \frac{3\pi}{2} - \alpha ) \sec( \alpha - \frac{5\pi}{2} ) + \tan( \frac{5\pi}{2} + \alpha ) \tan( \alpha - \frac{3\pi}{2} )$

$\sf =( - \cosec( \alpha )) \sec(2\pi + \frac{\pi}{2} - \alpha ) + \tan( 2\pi + \frac{\pi}{2} + \alpha ) \tan( \frac{3\pi}{2} - \alpha )$

$\sf =( - \cosec( \alpha )) \sec( \frac{\pi}{2} - \alpha ) + \tan( \frac{\pi}{2} + \alpha )( - \cot( \alpha ) )$

$\sf =( - \cosec( \alpha )) \times \cosec( \alpha ) + \cot( \alpha ) \times ( - \cot( \alpha ) )$

$\sf =- \cosec ^{2} ( \alpha ) + \cot ^{2} ( \alpha )$

$\sf =- 1(\cosec ^{2} ( \alpha ) - \cot ^{2} ( \alpha ) )$

$\sf = - 1 \times 1$

$\sf = - 1$

Hence proved.

Answer 2

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