Question:
$\tan ^{-1}\left(\tan \frac{2 \pi}{3}\right)$ is equal to __________________.
Solution:
$\tan ^{-1}\left(\tan \frac{2 \pi}{3}\right)$
$=\tan ^{-1}\left[\tan \left(\pi-\frac{\pi}{3}\right)\right]$
$=\tan ^{-1}\left(-\tan \frac{\pi}{3}\right)$
$=\tan ^{-1}\left[\tan \left(-\frac{\pi}{3}\right)\right]$ $[\tan (-\theta)=-\tan \theta]$
$=-\frac{\pi}{3}$ $\left[\tan ^{-1}(\tan x)=x\right.$, if $\left.x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\right]$
$\tan ^{-1}\left(\tan \frac{2 \pi}{3}\right)$ is equal to $-\frac{\pi}{3}$