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

Let $y={\cot }^{-1}\left(-\frac{1}{\sqrt{3}}\right)$.Using the property ${\cot }^{-1}(-x)=\pi -{\cot }^{-1}(x)$:
$$y=\pi -{\cot }^{-1}\left(\frac{1}{\sqrt{3}}\right)$$
We know that $\cot \left(\frac{\pi }{3}\right)=\frac{1}{\sqrt{3}}$, so ${\cot }^{-1}\left(\frac{1}{\sqrt{3}}\right)=\frac{\pi }{3}$.
$$y=\pi -\frac{\pi }{3}=\frac{2\pi }{3}$$
Since the range of the principal value of ${\cot }^{-1}x$ is $(0,\pi )$, and $\frac{2\pi }{3}\in (0,\pi )$, the principal value is $\frac{2\pi }{3}$.