Compute the given expression: $\;$ $\cos^{-1} \left[- \cos \left(\dfrac{3 \pi}{4}\right)\right]$
Given expression: $\;$ $\cos^{-1} \left[- \cos \left(\dfrac{3 \pi}{4}\right)\right]$
$= \pi - \cos^{-1} \left[\cos \left(\dfrac{3 \pi}{4}\right)\right]$ $\;\;\;$ $\left\{\because \;\; \cos^{-1} \left(- x\right) = \pi - \cos^{-1} x \;\; \forall \;\; x \in \left[-1, 1\right]\right\}$
$= \pi - \dfrac{3 \pi}{4}$ $\;\;\;$ $\left\{\because \;\; \cos^{-1} \left(\cos x\right) = x \;\; \forall \;\; x \in \left[0, \pi\right]\right\}$
$= \dfrac{\pi}{4}$