Solution

We know that:

\[ \tan^{-1}(\tan x) = x,\; x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \] \[ \cos^{-1}(\cos x) = x,\; x \in [0,\pi] \]

Therefore,

\[ \tan^{-1}\left(\tan \frac{5\pi}{6}\right) + \cos^{-1}\left(\cos \frac{13\pi}{6}\right) \]

\[ = \tan^{-1}\left(\tan\left(\pi – \frac{\pi}{6}\right)\right) + \cos^{-1}\left(\cos\left(2\pi + \frac{\pi}{6}\right)\right) \]

\[ = \tan^{-1}\left(-\tan\frac{\pi}{6}\right) + \cos^{-1}\left(\cos\frac{\pi}{6}\right) \]

Using identity: \[ \tan^{-1}(-x) = -\tan^{-1}x \]

\[ = -\frac{\pi}{6} + \frac{\pi}{6} \]

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