Compute the given expression: $\;$ $\tan \left[5 \tan^{-1} \left(\dfrac{\sqrt{3}}{3}\right) - \dfrac{1}{4} \sin^{-1} \left(\dfrac{\sqrt{3}}{2}\right)\right]$
Given expression: $\;$ $\tan \left[5 \tan^{-1} \left(\dfrac{\sqrt{3}}{3}\right) - \dfrac{1}{4} \sin^{-1} \left(\dfrac{\sqrt{3}}{2}\right)\right]$
$= \tan \left[5 \tan^{-1} \left(\dfrac{1}{\sqrt{3}}\right) - \dfrac{1}{4} \sin^{-1} \left(\dfrac{\sqrt{3}}{2}\right)\right]$
$= \tan \left[5 \times \dfrac{\pi}{6} - \dfrac{1}{4} \times \dfrac{\pi}{3}\right]$
$= \tan \left[\dfrac{9 \pi}{12}\right]$
$= \tan \left[\dfrac{3 \pi}{4}\right]$
$= \tan \left[\dfrac{\pi}{2} + \dfrac{\pi}{4}\right]$
$= - \cot \left[\dfrac{\pi}{4}\right]$
$= -1$