Explanation

(b)
Given, $\tan{\left\{ \sec^{- 1}\left( \frac{1}{x} \right) \right\} = \sin{(\tan^{- 1}2)}}$
$\Rightarrow \tan{\left( \tan^{- 1}\frac{\sqrt{1 - x^{2}}}{x} \right) = \sin\left( \sin^{- 1}\frac{2}{\sqrt{1 + 2^{2}}} \right)}$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left\lbrack \because\tan^{- 1}{x = \sin^{- 1}\frac{x}{\sqrt{1 + x^{2}}}} \right\rbrack$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{\sqrt{1 - x^{2}}}{x} = \frac{2}{\sqrt{5}}$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4x^{2} = 5(1 - x^{2})$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x^{2} = \frac{5}{9}\ \ \ \Rightarrow \ \ x = \frac{\sqrt{5}}{3}$