Explanation

(a)
Given, $\tan^{- 1}{x + \cos^{- 1}{\frac{y}{\sqrt{1 + y^{2}}} = \sin^{- 1}\frac{3}{\sqrt{10}}}}$
$\Rightarrow \ \ \tan^{- 1}{x + \tan^{- 1}{\frac{1}{y} = \sin^{- 1}\frac{3}{\sqrt{10}}}}$
$\Rightarrow \ \tan^{- 1}\left( \frac{x + \frac{1}{y}}{1 - \frac{x}{y}} \right) = \tan^{- 1}3$
$\Rightarrow \ \ x + \frac{1}{y} = 3\left( 1 - \frac{x}{y} \right)$
$\Rightarrow \ \ \ x = 1,\ \ \ y = 2$
$\therefore$ The number of solutions of given equation is 1.