the correct Option is OPTION (B).
Reason:
$\tan ^{-1} \sqrt{3}-\cot ^{1}(-\sqrt{3})$ can be written as
$=\tan ^{-1} \tan \frac{\pi}{3}-\cot ^{-1}\left(-\cot \frac{\pi}{6}\right)$
$=\frac{\pi}{3}-\cot ^{-1}\left[\cot \left(\pi-\frac{\pi}{6}\right)\right]$
$=\frac{\pi}{3}-\left(\pi-\frac{\pi}{6}\right)$
$=\frac{\pi}{3}-\frac{5 \pi}{6}$
$=\frac{-3 \pi}{6}$
$=-\pi / 2$