Question

Simplify the function: ${\cot }^{-1}\left(\sqrt{\frac{1+\cos x}{1-\cos x}}\right)$

Answer 1

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