Find the value of: $\;$ $\tan \left\{\sin^{-1} \left(\dfrac{3}{5}\right) + \cos^{-1} \left(\dfrac{5}{13}\right)\right\}$
$\left[\text{Note: } \sin^{-1} \left(x\right) = \tan^{-1} \left(\dfrac{x}{\sqrt{1 - x^2}}\right), \;\;\; 0 < x < 1 \right.$
$\left. \cos^{-1} \left(x\right) = \tan^{-1} \left(\dfrac{\sqrt{1 - x^2}}{x}\right), \;\;\; 0 < x < 1 \right.$
$\left. \tan^{-1} \left(x\right) + \tan^{-1} \left(y\right) = \pi + \tan^{-1} \left(\dfrac{x + y}{1 - x y}\right), \;\;\; x \times y > 1 \right]$
$\tan \left\{\sin^{-1} \left(\dfrac{3}{5}\right) + \cos^{-1} \left(\dfrac{5}{13}\right)\right\}$
$= \tan \left\{\tan^{-1} \left(\dfrac{\dfrac{3}{5}}{\sqrt{1 - \dfrac{9}{25}}}\right) + \tan^{-1} \left(\dfrac{\sqrt{1 - \dfrac{25}{169}}}{\dfrac{5}{13}}\right) \right\} $
$ = \tan \left\{\tan^{-1} \left(\dfrac{3}{4}\right) + \tan^{-1} \left(\dfrac{12}{5}\right) \right\} $
$ = \tan \left\{\pi + \tan^{-1} \left(\dfrac{\dfrac{3}{4} + \dfrac{12}{5}}{1 - \dfrac{3}{4} \times \dfrac{12}{5}}\right) \right\} $
$ = \tan \left\{\pi + \tan^{-1} \left(\dfrac{-63}{16}\right) \right\} $
$ = \tan \left\{\tan^{-1} \left(\dfrac{- 63}{16}\right) \right\} $
$ = - \dfrac{63}{16} $