Question

Show that sin (cot^-1 √1-cosx/ 1+cosx)=cos(x/2)

Answer 1

$\sin( \cot ^{ - 1} ( \sqrt{ \frac{1 - \cos(x) }{1 + \cos(x) } } ) )$

$\because \sin( \cot^{ - 1} ( \theta) ) = \frac{1}{ \sqrt{1 + { \theta}^{2} } }$

$= \frac{1}{ \sqrt{ 1 + \frac{1 + \cos(x)}{1 + \cos(x)}} }$

$= \sqrt{ \frac{1 + \cos( x ) }{2} }$

$= | \cos( \frac{x}{2} ) |$