Solution

We have:

\[ \tan^{-1}\left(\tan \frac{2\pi}{3}\right) = \tan^{-1}\left(\tan\left(\pi – \frac{\pi}{3}\right)\right) \]

\[ = \tan^{-1}\left(-\tan\frac{\pi}{3}\right) \]

\[ = -\tan^{-1}\left(\tan\frac{\pi}{3}\right) = -\frac{\pi}{3} \]

Note:
Remember that, \[ \tan^{-1}\left(\tan \frac{2\pi}{3}\right) \ne \frac{2\pi}{3} \]

Since, \[ \tan^{-1}(\tan x) = x,\; x \in \left(-\frac{\pi}{2},\frac{\pi}{2}\right) \]

and \[ \frac{2\pi}{3} \notin \left(-\frac{\pi}{2},\frac{\pi}{2}\right) \]

\[ \boxed{-\frac{\pi}{3}} \]