Solution:

$\tan ^{-1} \sqrt{3}-\cot ^{-1}(-\sqrt{3})$
Putting the value of $\tan ^{-1} \sqrt{3}$ and using the formula
$\begin{array}{l}
\cot ^{-1}(-x)=\pi-\cot ^{-1} x \\
=\frac{\pi}{3}-\left(\pi-\cot ^{-1}(\sqrt{3})\right)
\end{array}$
Putting the value of $\cot ^{-1}(\sqrt{3})$
$\begin{array}{l}
=\frac{\pi}{3}-\left(\pi-\frac{\pi}{6}\right) \\
=\frac{\pi}{3}-\frac{5 \pi}{6} \\
=-\frac{3 \pi}{6} \\
=-\frac{\pi}{2}
\end{array}$