The principal value of $\cot ^{-1}\left(-\frac{1}{\sqrt{3}}\right)$

$\frac{\pi}{3}$ $-\frac{\pi}{6}$ $\frac{1}{\sqrt{3}}$ $\frac{2 \pi}{3}$

$\frac{2 \pi}{3}$

$\cot ^{-1}\left(\frac{-1}{\sqrt{3}}\right)=\pi-\cot ^{-1}\left(\frac{1}{\sqrt{3}}\right)$ as $\cot^{-1}(-x)$ $\pi - \cot^{-1}(x)$ $=\pi-\frac{\pi}{3}=\frac{2 \pi}{3}$