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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 $-\frac{\pi}{3}$

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