Explanation

(d)
$m\tan{(\theta - 30{^\circ}) = n\tan{(\theta + 120{^\circ})}}$
$\Rightarrow \ \ m\left( \frac{\tan{\theta - \frac{1}{\sqrt{3}}}}{1 + \frac{1}{\sqrt{3}}\tan\theta} \right) = - n\left( \frac{- \tan{\theta + \sqrt{3}}}{1 + \sqrt{3}\tan\theta} \right)$
$\Rightarrow \ \ m\left\lbrack \left( \sqrt{3}\tan\theta \right)^{2} - 1 \right\rbrack = - n( - \tan^{2}{\theta + 3})$
$\Rightarrow \ \ 3m\tan^{2}\theta - m = n\tan^{2}{\theta - 3n}$
$\Rightarrow \ \tan^{2}{\theta = \frac{m - 3n}{3m - n}}$
Now, $\cos{2\theta = \frac{1 - \tan^{2}\theta}{1 + \tan^{2}\theta}}$
$= \frac{1 - \frac{m - 3n}{3m - n}}{1 + \frac{m - 3n}{3m - n}} = \frac{3m - n - m + 3n}{3m - n + m - 3n}$
$= \frac{2(m + n)}{4(m - n)} = \frac{m + n}{2(m - n)}$