Explanation

(d)
We have, $x + \frac{1}{x} = 2\cos\theta$
$\Rightarrow \ \ \left( x + \frac{1}{x} \right)^{3} = \left( 2\cos\theta \right)^{3}$
$\Rightarrow \ \ x^{3} + \frac{1}{x^{3}} + 3x.\frac{1}{x}\left( x + \frac{1}{x} \right) = 8\cos^{3}\theta$
$\Rightarrow \ \ x^{3} + \frac{1}{x^{3}} + 3.2\cos{\theta = 8\cos^{3}\theta}$
$\Rightarrow \ \ x^{3} + \frac{1}{x^{3}} = 2(4\cos^{3}{\theta - 3\cos\theta})$
$= 2\cos{3\theta}$