sum of all Distinct solution of the equation $ \sqrt{3}\sec x+\csc x+2(\tan x-\cot x) = 0\;,$ 
The sum of all Distinct solution of the equation $\displaystyle \sqrt{3}\sec x+\csc x+2(\tan x-\cot x) = 0\;,$
Where $x\in (-\pi,\pi)$ and $\displaystyle x\neq 0,\neq \frac{\pi}{2}.$

$\bf{My\; Try::}$ We can write equation as $$\frac{\sqrt{3}\sin x+\cos x}{\sin x\cos x}-2\frac{\cos 2x}{\sin x\cos x} = 0$$
So we get $$2\frac{\cos \left(x+\frac{\pi}{3}\right)}{\sin x\cos x}-2\frac{\cos 2x}{\sin x\cos x}=0$$
So we get $$\cos (2x) = \cos\left(x+\frac{\pi}{3}\right)$$
So we Get $\displaystyle 2x=2n\pi\pm (x+\frac{\pi}{3})\;,n\in \mathbb{Z}$
So we get $\displaystyle x= 2n\pi+\frac{\pi}{3}$ or $\displaystyle x = \frac{2n\pi-\frac{\pi}{3}}{3}\;,$ Where $n\in \mathbb{Z}$
So we get $$x=\frac{\pi}{3} = \frac{3\pi}{9}\;\;,-\frac{5\pi}{9}\;\;,-\frac{7\pi}{9}$$
So Sum of distinct Roots is $\displaystyle = -\pi$
But it is wrong, Where i have done wrong, Help me
Thanks

Observe that:$$\sqrt{3}\sin x+\cos x = 2\cdot \left(\frac{\sqrt{3}}{2}\sin x+\frac{1}{2}\cos x\right)$$ $$ = 2\cdot \left(\sin\frac{\pi}{3}\sin x+cos\frac{\pi}{3}\cos x\right)$$ $$=2\cos(x\color{red}{-}\frac{\pi}{3})$$
But instead of $2\cos(x\color{red}{-}\frac{\pi}{3})$, you have written $2\cos(x\color{red}{+}\frac{\pi}{3})$ in the following step and proceeded accordingly:

So we get $$2\frac{\cos \left(x+\frac{\pi}{3}\right)}{\sin x\cos x}-2\frac{\cos 2x}{\sin x\cos x}=0$$

