Explanation

(d)
Given, $\sin{\left\lbrack \cot^{- 1}(1 + x) \right\rbrack = \cos{(\tan^{- 1}x)}}$
$\therefore\ \sin\left( \sin^{- 1}\frac{1}{\sqrt{1 + (1 + x^{2})}} \right) = \cos\left( \cos^{- 1}\frac{1}{\sqrt{1 + x^{2}}} \right)$
$\Rightarrow \ \frac{1}{\sqrt{1 + (1 + x^{2})}} = \frac{1}{\sqrt{1 + x^{2}}}$
$\Rightarrow 1 + x^{2} + 2x + 1 = x^{2} + 1$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x = - \frac{1}{2}$