Let \(\mathrm{S}=\left\{x \in(-\pi, \pi) \mid x \neq 0, \pm \frac{\pi}{2}\right\}\). The sum of all distinct solutions of the equation \(\sqrt{3} \sec x+\operatorname{cosec} x+2(\tan x-\cot x)=0\) in the set S is equal to

\(\sqrt{3} \sec x+\operatorname{cosec} x+2(\tan x-\cot x)=0 \)
\( \therefore \frac{\sqrt{3}}{2} \sec x+\frac{1}{2} \operatorname{cosec} x=\cot x-\tan x \)
\( \therefore \frac{\sqrt{3}}{2} \times \frac{1}{\cos x}+\frac{1}{2} \times \frac{1}{\sin x}=\frac{\cos x}{\sin x}-\frac{\sin x}{\cos x} \)
\( \therefore \frac{\sqrt{3}}{2} \sin x+\frac{1}{2} \cos x=\cos ^2 x-\sin ^2 x \)
\( \therefore \cos \left(\frac{\pi}{3}-x\right)=\cos 2 x \)
\( \therefore \frac{\pi}{3}-x=2 \mathrm{n} \pi \pm 2 x\)
for \(\mathrm{n}=0: x=\frac{\pi}{9} \in(-\pi, \pi)\) or \(\frac{-\pi}{3} \in(-\pi, \pi)\) for \(\mathrm{h}=1: x=\frac{-5 \pi}{9} \in(-\pi, \pi)\) or \(x=\frac{5 \pi}{3} \notin(-\pi, \pi)\) for \(\mathrm{n}=-1: x=\frac{-7 \pi}{3} \notin(-\pi, \pi)\) or \(x=\frac{7 \pi}{9} \in(-\pi, \pi)\) for \(n=2: x=\frac{11 \pi}{3} \notin(-\pi, \pi)\) or \(x=\frac{-11 \pi}{9} \notin(-\pi, \pi)\)
\(\therefore\) Distinct solutions are : \(\frac{\pi}{9}, \frac{-\pi}{3}, \frac{-5 \pi}{9}, \frac{7 \pi}{9}\)
\(\therefore\) Required sum \(=\frac{\pi-3 \pi-5 \pi+7 \pi}{9}=0\)