Explanation

(d)
Given, $\cos{x = 3\cos y}$
$\Rightarrow \ \ \frac{3}{1} = \frac{\cos x}{\cos y}$
Applying componendo and dividend, we get
$\frac{3 + 1}{3 - 1} = \frac{\cos{x + \cos y}}{\cos{x - \cos y}}$
$\Rightarrow \ \ 2 = \frac{2\cos{\left( \frac{x + y}{2} \right)\cos\left( \frac{x - y}{2} \right)}}{2\sin{\left( \frac{x + y}{2} \right)\sin\left( \frac{y - x}{2} \right)}}$
$\Rightarrow \ \ 2 = \cot\left( \frac{x + y}{2} \right)\cot\left( \frac{y - x}{2} \right)$
$\Rightarrow \ 2\tan\left( \frac{y - x}{2} \right) = \cot\left( \frac{x + y}{2} \right)$