Compute the given expression: $\;$ $\cot \left\{\dfrac{1}{2} \cos^{-1} \left(\dfrac{-4}{7}\right)\right\}$
Given expression: $\;$ $\cot \left\{\dfrac{1}{2} \cos^{-1} \left(\dfrac{-4}{7}\right)\right\}$
$= \cot \left\{\dfrac{1}{2} \left[\pi - \cos^{-1} \left(\dfrac{4}{7}\right)\right]\right\}$ $\;\;\;$ $\left\{\because \; \cos^{-1} \left(-x\right) = \pi - \cos^{-1} x \;\; \forall \;\; x \in \left[-1, 1\right]\right\}$
$= \cot \left\{\dfrac{\pi}{2} - \dfrac{1}{2} \cos^{-1} \left(\dfrac{4}{7}\right)\right\}$
$= \tan \left\{\dfrac{1}{2} \cos^{-1} \left(\dfrac{4}{7}\right)\right\}$ $\;\;\;$ $\left\{\because \; \cot \left(\dfrac{\pi}{2} - \theta\right) = \tan \theta\right\}$
$= \sqrt{\dfrac{1 - \cos \left\{\cos^{-1} \left(\dfrac{4}{7}\right)\right\}}{1 + \cos \left\{\cos^{-1} \left(\dfrac{4}{7}\right)\right\}}}$ $\;\;\;$ $\left\{\because \; \tan \left(\dfrac{\theta}{2}\right) = \sqrt{\dfrac{1 - \cos \theta}{1 + \cos \theta}}\right\}$
$= \sqrt{\dfrac{1 - \dfrac{4}{7}}{1 + \dfrac{4}{7}}}$ $\;\;\;$ $\left\{\because \; \cos \left(\cos^{-1} x \right) = x \;\; \forall \;\; x \in \left[-1, 1\right]\right\}$
$= \sqrt{\dfrac{3}{11}}$