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

Let the principle value be given by $x$

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

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

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

$\Rightarrow \tan x=\tan \left(-\frac{\pi}{3}\right)(\because-\tan (\theta)=\tan (-\theta))$

$\Rightarrow x=-\frac{\pi}{3}$