The value of $tan^{-1}(tan(-6))$, is

$2\pi - 6 $

We know that $tan^{-1} (tan \theta)= \theta $, if $-\frac{\pi}{2}< \theta < \frac{\pi}{2}$.

Here, $ \theta = - 6$ radians which does not lie between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$.

However, $2\pi - 6 $ lies between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ such that 

$tan (2\pi - 6 ) = - tan 6 = tan (-6)$

$∴ tan^{-1}[tan (-6)]= tan^{-1} [tan (2\pi - 6)] = 2 \pi - 6 $