${\tan ^{ - 1}}\left( {\sqrt {\frac{{1 - \cos x}}{{1 + \cos x}}} } \right),\;x < \pi$

${\tan ^{ - 1}}\left( {\sqrt {\frac{{1 - \cos x}}{{1 + \cos x}}} } \right) = {\tan ^{ - 1}}\left( {\sqrt {\frac{{2{{\sin }^2}(x/2)}}{{2{{\cos }^2}(x/2)}}} } \right)$
$= {\tan ^{ - 1}}\left( {\tan \frac{x}{2}} \right) = \frac{x}{2}$