Write the function in the simplest form:

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

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

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

$=\frac{x}{2}$