Find the value of: $\;$ $\cot \left[\text{cosec}^{-1} \left(\dfrac{5}{3}\right) + \tan^{-1} \left(\dfrac{2}{3}\right)\right]$
$\left[\text{Note: } \text{cosec}^{-1} \left(x\right) = \sin^{-1} \left(\dfrac{1}{x}\right) \right.$
$\left. \sin^{-1} \left(x\right) = \tan^{-1} \left(\dfrac{x}{\sqrt{1 - x^2}}\right), \;\;\; 0 < x < 1 \right.$
$\left. \tan^{-1} \left(x\right) + \tan^{-1} \left(y\right) = \tan^{-1} \left(\dfrac{x + y}{1 - x y}\right), \;\;\; x y < 1 \right.$
$\left. \tan^{-1} \left(x\right) = \cot^{-1} \left(\dfrac{1}{x}\right) \right]$
$\begin{aligned}
\cot \left[\text{cosec}^{-1} \left(\dfrac{5}{3}\right) + \tan^{-1} \left(\dfrac{2}{3}\right)\right] & = \cot \left[\sin^{-1} \left(\dfrac{3}{5}\right) + \tan^{-1} \left(\dfrac{2}{3}\right)\right] \\\\
& = \cot \left[\tan^{-1} \left(\dfrac{\dfrac{3}{5}}{\sqrt{1 - \dfrac{9}{25}}}\right) + \tan^{-1} \left(\dfrac{2}{3}\right)\right] \\\\
& = \cot \left[\tan^{-1} \left(\dfrac{3}{4}\right) + \tan^{-1} \left(\dfrac{2}{3}\right)\right] \\\\
& = \cot \left[\tan^{-1} \left(\dfrac{\dfrac{3}{4} + \dfrac{2}{3}}{1 - \dfrac{3}{4} \times \dfrac{2}{3}}\right)\right] \\\\
& = \cot \left[\tan^{-1} \left(\dfrac{17}{6}\right)\right] \\\\
& = \cot \left[\cot^{-1} \left(\dfrac{6}{17}\right)\right] \\\\
& = \dfrac{6}{17}
\end{aligned}$