Question

Solve this problem must

Answer 1

$\tan^{ - 1} ( \frac{\cos(x)}{1 + \sin(x) }) \\ we \: know \: that \: \\ \cos(x) = \cos ^{2} ( \frac{x}{2} ) - \sin ^{2} ( \frac{x}{2} ) \\ 1 + \sin(x) = \: ( \sin( \frac{x}{2} ) + \cos( \frac{x}{2} ) \: ) ^{2} \\ Now Plugin \: these \: values .. we \: get \\ \tan^{ - 1} ( \frac{\cos ^{2} ( \frac{x}{2} ) - \sin ^{2} ( \frac{x}{2} ) }{\: ( \sin( \frac{x}{2} ) + \cos( \frac{x}{2} ) \: ) ^{2} }$
$\tan^{ - 1}( \frac{ \cos( \frac{x}{2}) - \sin( \frac{x}{2} ) ) }{ \cos( \frac{x}{2} )+ \sin( \frac{x}{2} ) )}$
$divide \: the \: numerator \: and \: Denominator \: by \: \cos( \frac{x}{2} )$
$= > \tan^{ - 1}( \frac{1 - \tan( \frac{x}{2} ) }{1 + \tan( \frac{x}{2} ) } )$
$= \: \tan^{ - 1}( \tan( \frac{\pi}{4} - \frac{x}{2} ))$

$= > \frac{\pi}{4} - \frac{x}{2}$