Explanation

(d)
Let $\cot^{- 1}{x = \theta \Rightarrow x = \cot\theta}$
Now, $cosec\ \theta = \sqrt{1 + \cot^{2}\theta =}\sqrt{1 + x^{2}}$
$\Rightarrow \sin{\theta = \frac{1}{cosec\ \theta} = \frac{1}{\sqrt{1 + x^{2}}}}$
$\Rightarrow \theta = \sin^{- 1}\frac{1}{\sqrt{1 + x^{2}}}$
$\therefore\sin{(\cot^{- 1}{x) = \sin\left( \sin^{- 1}\ \frac{1}{\sqrt{1 + x^{2}}} \right)}}$
$= \frac{1}{\sqrt{\ }1 + x^{2}} = \left( 1 + x^{2} \right)^{- 1/2}$