The general solution of the equation $\frac{\tan x}{\tan 2x}+\frac{\tan 2x}{\tan x}+2=0$ is

$\frac{\tan x}{\tan 2x}+\frac{\tan 2x}{\tan x}+2=0$

${\tan }^{2}x+{\tan }^{2}2x+2\tan x.\tan 2x=0$

$\tan x,\tan 2x\ne 0$

${(\tan x+\tan 2x)}^2=0$

$\tan x+\tan 2x=0\Rightarrow \frac{\sin 3x}{\cos x\cos 2x}=0$

$\Rightarrow \sin 3x=0\Rightarrow n\pi =3xn\in Z$

$x=\frac{n\pi }{3}$