Find the value of: $\;$ $\cot \left[\dfrac{\pi}{4} - 2 \cot^{-1} \left(3\right)\right]$
$\begin{aligned} \cot \left[\dfrac{\pi}{4} - 2 \cot^{-1} \left(3\right)\right] & = \cot \left[\dfrac{\pi}{4} - 2 \tan^{-1} \left(\dfrac{1}{3}\right)\right] \\\\ & \left[\text{Note: } 2 \tan^{-1} \left(x\right) = \tan^{-1} \left(\dfrac{2 x}{1 - x^2}\right), \;\; \text{if } \; x^2 < 1\right] \\\\ & = \cot \left[\dfrac{\pi}{4} - \tan^{-1} \left(\dfrac{\dfrac{2}{3}}{1 - \dfrac{1}{9}}\right)\right] \\\\ & = \cot \left[\dfrac{\pi}{4} - \tan^{-1} \left(\dfrac{3}{4}\right)\right] \\\\ & = \cot \left[\dfrac{\pi}{4} - \cot^{-1} \left(\dfrac{4}{3}\right)\right] \\\\ & \left[\text{Note: } \cot \left(\alpha - \beta\right) = \dfrac{\cot \alpha \times \cot \beta + 1}{\cot \beta - \cot \alpha}\right] \\\\ & = \dfrac{\cot \left(\dfrac{\pi}{4}\right) \times \cot \left[\cot^{-1} \left(\dfrac{4}{3}\right)\right] + 1}{\cot \left[\cot^{-1} \left(\dfrac{4}{3}\right)\right] - \cot \left(\dfrac{\pi}{4}\right)} \\\\ & = \dfrac{\dfrac{4}{3} + 1}{\dfrac{4}{3} - 1} \\\\ & = 7 \end{aligned}$