Question

Let $S=\{x\in (-\pi ,\pi ):x\ne 0,\pm \frac{\pi }{2}\}$. The sum of all distinct solutions of the equation $\sqrt{3}\sec x+\text{cosec}x+2(\tan x-\cot x)=0$ in the set $S$ is equal to:

• A. $-\frac{7\pi }{9}$
• B. $-\frac{2\pi }{9}$
• C. $0$
• D. $\frac{5\pi }{9}$

Answer 1

The correct option is C $0$

$\frac{\sqrt{3}}{\cos x}+\frac{1}{\sin x}+2(\frac{\sin x}{\cos x}-\frac{\cos x}{\sin x})=0$
$\Rightarrow \cos 2x=\cos (x-\frac{\pi }{3})$
$\Rightarrow 2x=2n\pi \pm (x-\frac{\pi }{3})$

$x=2n\pi -\frac{\pi }{3}\text{or}x=\frac{2n\pi }{3}+\frac{\pi }{9}$
$\Rightarrow x=-\frac{\pi }{3}\text{or}x=\frac{\pi }{9},\frac{7\pi }{9},-\frac{5\pi }{9}$

Sum$=\frac{-3\pi +\pi +7\pi -5\pi }{9}=0$