Question

The number of common solution(s) of the trigonometric equations $\cos 2x+(1-\sqrt{3})=(2-\sqrt{3})\cos x$ and $\sin 3x=2\sin x,$ satisfying the inequality $\sqrt{3}\tan x-1\ge 0$ in $[0,5\pi ]$ is

Answer 1

$2{\cos }^{2}x-1+1-\sqrt{3}=(2-\sqrt{3})\cos x$
$\Rightarrow \cos x(\cos x-1)+\sqrt{3}(\cos x-1)=0$
$\Rightarrow (\cos x-1)(2\cos x+\sqrt{3})=0$
$\Rightarrow \cos x=1$ or $\cos x=-\frac{\sqrt{3}}{2}$

$\sin 3x=2\sin x$
$\Rightarrow 3\sin x-4{\sin }^{3}x=2\sin x$
$\Rightarrow \sin x=0$ or $3-4{\sin }^{2}x=2$

$4{\sin }^{2}x=1$
$\Rightarrow \sin x=\pm \frac{1}{2}$

Common solutions in $[0,5\pi ]$ are $0,2\pi ,4\pi ,\frac{5\pi }{6},\frac{7\pi }{6},\frac{17\pi }{6},\frac{19\pi }{6},\frac{29\pi }{6}$

If $\tan x\ge \frac{1}{\sqrt{3}}$
$\Rightarrow x\in [\frac{\pi }{6}+n\pi ,\frac{\pi }{2}+n\pi ];n\in Z$

So, common solutions $x=\frac{7\pi }{6},\frac{19\pi }{6}$