Find the principal value of each of the following :

$\tan ^{-1} \sqrt{3}-\cot ^{-1}(-\sqrt{3})$

Putting the value of $\tan ^{-1} \sqrt{3}$ and using the formula

$\cot ^{-1}(-x)=\pi-\cot ^{-1} x$

$=\frac{\pi}{3}-\left(\pi-\cot ^{-1}(\sqrt{3})\right)$

Putting the value of $\cot ^{-1}(\sqrt{3})$

$=\frac{\pi}{3}-\left(\pi-\frac{\pi}{6}\right)$

$=\frac{\pi}{3}-\frac{5 \pi}{6}$

$=-\frac{3 \pi}{6}$

$=-\frac{\pi}{2}$