Mathematics - Trigonometric Equation Question with Solution | TestHub

$\frac{\sqrt{3}}{\mathrm{Cosx}}+\frac{1}{\mathrm{Sinx}}+2\mathrm{}\left(\frac{\mathrm{Sinx}}{\mathrm{cosx}}-\frac{\mathrm{Cosx}}{\mathrm{sinx}}\right)=0\mathrm{}$
$\sqrt{3}\sin x+\cos x=2\cos 2x$
$\Rightarrow$$\cos 2x=\cos \left(x-\frac{\pi }{3}\right)$
$\Rightarrow 2x=2n\pi \pm \left(x-\frac{\pi }{3}\right)$,$\left\{\begin{array}{l}2x=2n\pi +x-\frac{\pi }{3}\\ 2x=2n\pi -x+\frac{\pi }{3}\end{array}\right.$
$\mathrm{\Rightarrow x}=\left(6n-1\right)\frac{\pi }{3}$or$\left(6n+1\right)\frac{\pi }{9}$
$\Rightarrow$$x= -\frac{\pi }{3},\frac{\pi }{9},\frac{7\pi }{9}$and$-\frac{5\pi }{9}$in$\left(-\pi , \pi \right)$
$\Rightarrow sum = 0$