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To Find: The Principle value of $\cot ^{-1}(-\sqrt{3})$

Let the principle value be given by $x$

Now, let $x=\cot ^{-1}(-\sqrt{3})$

$\Rightarrow \cot x=-\sqrt{3}$

$\Rightarrow \cot x=-\cot \left(\frac{\pi}{6}\right)\left(\because \cot \left(\frac{\pi}{6}\right)=\sqrt{3}\right)$

$\Rightarrow \cot x=\cot \left(\pi-\frac{\pi}{6}\right)(\because-\cot (\theta)=\cot (\pi-\theta))$

$\Rightarrow x=\frac{5 \pi}{6}$