Explanation

(a)
Given, $\cot{\left( \cos^{- 1}x \right) = \sec\left( \tan^{- 1}\frac{a}{\sqrt{b^{2} - a^{2}}} \right)}$
$\therefore\ \ \ \cot\left( \cot^{- 1}\left( \frac{x}{\sqrt{1 - x^{2}}} \right) \right)$
$\ \ \ \ \ \ \ \ \ \ \ \ \ = \sec\left( \sec^{- 1}\frac{b}{\sqrt{b^{2} - a^{2}}} \right)$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{x}{\sqrt{1 - x^{2}}} = \frac{b}{\sqrt{b^{2} - a^{2}}}$
$\Rightarrow \ \ \ \ \ x^{2}\left( b^{2} - a^{2} \right) = b^{2} - b^{2}x^{2}$
$\Rightarrow \ \ x^{2}\left( 2b^{2} - a^{2} \right) = b^{2}$
$\Rightarrow \ \ \ x = \frac{b}{\sqrt{2b^{2} - a^{2}}}$