Explanation

(c)
Let $y_{1} = \cos^{- 1}\left( \frac{1 - x^{2}}{1 + x^{2}} \right)$
and $y_{2} = \cot^{- 1}\left( \frac{1 - 3x^{2}}{3x - x^{3}} \right)$
Put $x = \tan{\theta \Rightarrow \theta = \tan^{- 1}x}$
$\Rightarrow y_{1} = 2\tan^{- 1}{x\ and\ y_{2} = 3\tan^{- 1}x}$
On differentiating w.r.t. $x$, we get
$\frac{dy_{1}}{dx} = \frac{2}{1 + x^{2}}\ and\ \frac{dy_{2}}{dx} = \frac{3}{1 + x^{2}}$
$\Rightarrow \frac{dy_{1}}{dy_{2}} = \frac{\left( \frac{dy_{1}}{dx} \right)}{\left( \frac{dy_{2}}{dx} \right)} = \frac{\left( \frac{2}{1 + x^{2}} \right)}{\left( \frac{3}{1 + x^{2}} \right)} = \frac{2}{3}$