Question

Number of solutions of the equation $2\sin x-2\sqrt{3}\cos x-\sqrt{3}\tan x+3=0$ where $x\in [0,2\pi )$ is

Answer 1

$2\sin x-2\sqrt{3}\cos x-\sqrt{3}\tan x+3=0$
$\Rightarrow 2[\sin x-\sqrt{3}\cos x]-\frac{\sqrt{3}}{\cos x}[\sin x-\sqrt{3}\cos x]=0$
$\Rightarrow [\sin x-\sqrt{3}\cos x][2-\frac{\sqrt{3}}{\cos x}]=0$
For $\sin x-\sqrt{3}\cos x=0$
$\Rightarrow \tan x=\sqrt{3}$
$x=\frac{\pi }{3},\frac{4\pi }{3}$
For $2-\frac{\sqrt{3}}{\cos x}=0$
$\Rightarrow \cos x=\frac{\sqrt{3}}{2}$
$x=\frac{\pi }{6},\frac{11\pi }{6}$
So, $x=\frac{\pi }{6},\frac{\pi }{3},\frac{4\pi }{3},\frac{11\pi }{6}$