Find the principal value of $\tan^{-1} \left(\sqrt{3}\right)$
Let $\;$ $\tan^{-1} \left(\sqrt{3}\right) = y$, $\;$ where $\;$ $- \dfrac{\pi}{2} < y < \dfrac{\pi}{2}$
$\implies$ $\tan \left(y\right) = \sqrt{3} = \tan \left(\dfrac{\pi}{3}\right)$
$\implies$ $y = \dfrac{\pi}{3}$
$\therefore \;$ The principal value of $\tan^{-1} \left(\sqrt{3}\right)$ is $\dfrac{\pi}{3}$