Question 1

Verify the Identity.

Question 2

Explanation:

$( \tan( \alpha ) )/(1 + \sec( \alpha ) ) = - \cot( \alpha ) + \csc( \alpha )$

Multiply by 1-sec(a)

$( \tan( \alpha ) (1 - \sec( \alpha ) )/((1 + \sec( \alpha ) )(1 - \sec( \alpha ) ))$

$\frac{ \tan( \alpha ) - \tan( \alpha ) \sec( \alpha ) }{1 - \sec {}^(2) ( \alpha ) }$

$\frac{ \tan( \alpha ) }{1 - \sec {}^(2) ( \\ \alpha ) } - \frac{ \tan( \alpha ) \sec( \alpha ) }{1 - \sec {}^(2) ( \alpha ) }$

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

$- (1)/( \tan( \alpha ) ) + ( \sec( \alpha ) )/( \tan( \alpha ) )$

$- \cot( \alpha ) + ( (1)/( \cos( \alpha ) ) )/( ( \sin( \alpha ) )/( \cos( \alpha ) ) )$

$- \cot( \alpha ) + (1)/( \sin( \alpha ) )$

$- \cot( \alpha ) + \csc( \alpha )$

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