Question

Let $S=\left\{xϵ\right(-\pi ,\pi ):x\ne 0,\pm \frac{\pi }{2}\}$. The sum of all distinct solutions of the equation $\sqrt{3}secx+cosecx+2(tanx-cotx)=0$ 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 $0$

$\sqrt{3}secx+cosecx+2(tanx-cotx)=0\Rightarrow \frac{\sqrt{3}}{2}sinx+\frac{1}{2}cosx=co{s}^{2}x-si{n}^{2}x\Rightarrow cos(x-\frac{\pi }{3})=cos2x\Rightarrow x-\frac{\pi }{3}=2n\pi \pm 2x\Rightarrow x=\frac{2n\pi }{3}+\frac{\pi }{9}orx=-2n\pi -\frac{\pi }{3}$
For $xϵS,n=0\Rightarrow x=\frac{\pi }{9},-\frac{\pi }{3}$
$n=1\Rightarrow x=\frac{7\pi }{9};n=-1\Rightarrow x=\frac{-5\pi }{9}$
$\therefore$ Sum of all values of $x=\frac{\pi }{9}-\frac{\pi }{3}+\frac{7\pi }{3}-\frac{5\pi }{9}=0$