Evaluate: $\;$ $\tan^{-1} \left[\tan \left(\dfrac{5 \pi}{4}\right)\right]$
$\because \;$ $\dfrac{5 \pi}{4} \notin \left(- \dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$
$\begin{aligned}
\therefore \; \tan^{-1} \left[\tan \left(\dfrac{5 \pi}{4}\right)\right] & = \tan^{-1} \left[\tan \left(\pi - \dfrac{5 \pi}{4}\right)\right] \\\\
& = \tan^{-1} \left[- \tan \left(- \dfrac{\pi}{4}\right)\right] \hspace{1cm} \left[\text{Note: } \tan \left(\pi - \theta\right) = - \tan \theta\right] \\\\
& = \tan^{-1} \left[\tan \left(\dfrac{\pi}{4}\right)\right] \hspace{1cm} \left[\text{Note: } \tan \left(- \theta\right) = - \tan \theta\right] \\\\
& = \dfrac{\pi}{4} \hspace{1cm} \left[\dfrac{\pi}{4} \in \left(- \dfrac{\pi}{2}, \dfrac{\pi}{2}\right)\right]
\end{aligned}$