Question

Prove that
$\frac{ \sec( \alpha ) - \tan( \alpha ) }{ \cosec( \alpha ) + \cot( \alpha ) } = \frac{ \cosec( \alpha ) - \cot( \alpha ) }{ \cosec( \alpha ) + \tan( \alpha ) }$
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Answer 1

In the denominator it should be sec instead of cosec.

Question:

$\frac{ \sec( \alpha ) - \tan( \alpha ) }{ \csc( \alpha ) + \cot( \alpha ) } = \frac{ \csc( \alpha ) - \cot( \alpha ) }{ \sec( \alpha ) + \tan( \alpha ) }$

Solution:

$= \frac{ \sec( \alpha ) - \tan( \alpha ) }{ \csc( \alpha ) + \cot( \alpha ) } \times \frac{ \csc( \alpha ) - \cot( \alpha ) }{ \csc( \alpha ) - \cot( \alpha ) }$

$= \frac{( \sec( \alpha ) - \tan( \alpha ) )( \csc( \alpha ) - \cot( \alpha ) )}{ {csc}^{2}( \alpha ) - {cot}^{2} ( \alpha )}$

$= ( \sec( \alpha ) - \tan( \alpha ) )( \csc( \alpha ) - \cot( \alpha ) )$

$= ( \sec( \alpha ) - \tan( \alpha ) )( \csc( \alpha ) - \cot( \alpha ) ) \times \frac{( \sec( \alpha ) + \tan( \alpha ) ) }{ ( \sec( \alpha ) + \tan( \alpha ) ) }$

$= \frac{( \csc( \alpha ) - \cot( \alpha ) )( {sec}^{2}( \alpha ) - {tan}^{2} ( \alpha )}{( \ \sec( \alpha ) + \tan( \alpha ) ) }$

$= \frac{( \csc( \alpha ) - \cot( \alpha ) )}{( \sec( \alpha ) + \tan( \alpha )) }$

Hope it helps you..!! ❤️