Explanation

(d)
Given, $\cot{x + cosec\ x = \sqrt{3}}$
$\Rightarrow \ \ \frac{\cos{x + 1}}{\sin x} = \sqrt{3}$
$\Rightarrow \ \ \frac{2\cos^{2}\frac{x}{2}}{2\sin{\frac{x}{2}\cos\frac{x}{2}}} = \sqrt{3}\ \ \Rightarrow \ \tan{\frac{x}{2} = \frac{1}{\sqrt{3}}}$
$\therefore\ \ \ \frac{x}{2} = n\pi + \frac{\pi}{6}\ \ \ \Rightarrow \ \ \ x = 2n\pi + \frac{\pi}{3}$
$\Rightarrow \ \ x - \frac{\pi}{6} = 2n\pi + \frac{\pi}{6}$
For $n = 0$, principle value of $x - \frac{\pi}{6}$ is $\frac{\pi}{6}$