Question

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

Answer: (C)

$0$

$\sqrt{3}\sec x+cosecx+2(\tan x-cotx)=0\Rightarrow \sqrt{3}\sin x+\cos x+2({\sin }^{2}x-{\cos }^{2}x)=0$
$\sqrt{3}\sin x+\cos x-2\cos 2x=0\Rightarrow \sin (x+\frac{\pi }{3})=\cos 2x$
$\cos (\frac{\pi }{3}-x)=\cos 2x\Rightarrow 2x=2n\pi \pm (\frac{\pi }{3}-x)$
$x=\frac{2n\pi }{3}+\frac{\pi }{9}$ or $x=2n\pi -\frac{\pi }{3}$
$x=-{100}^o,-{60}^0,{20}^0,{140}^0$
s0 $-{100}^o-{60}^o+{20}^o+{140}^o=0$