Let $\mathrm{S}=\left\{x \in(-\pi, \pi) \mid x \neq 0, \pm \frac{\pi}{2}\right\}$. The sum of all distinct…
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
$-\frac{7 \pi}{9}$
$-\frac{2 \pi}{9}$
0
$\frac{5 \pi}{9}$
Solution
$\begin{aligned}
& \sqrt{3} \sec x+\operatorname{cosec} x+2(\tan x-\cot x)=0 \\
& \therefore \quad \frac{\sqrt{3}}{2} \sec x+\frac{1}{2} \operatorname{cosec} x=\cot x-\tan x \\
& \therefore \quad \frac{\sqrt{3}}{2} \times \frac{1}{\cos x}+\frac{1}{2} \times \frac{1}{\sin x}=\frac{\cos x}{\sin x}-\frac{\sin x}{\cos x} \\
& \therefore \quad \frac{\sqrt{3}}{2} \sin x+\frac{1}{2} \cos x=\cos ^2 x-\sin ^2 x \\
& \therefore \quad \cos \left(\frac{\pi}{3}-x\right)=\cos 2 x \\
& \therefore \quad \frac{\pi}{3}-x=2 \mathrm{n} \pi \pm 2 x
\end{aligned}$
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 $x=\frac{-11 \pi}{9} \notin(-\pi, \pi)$
$\therefore \quad$ Distinct solutions are : $\frac{\pi}{9}, \frac{-\pi}{3}, \frac{-5 \pi}{9}, \frac{7 \pi}{9}$
$\therefore \quad$ Required sum $=\frac{\pi-3 \pi-5 \pi+7 \pi}{9}=0$