Question

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

Answer 1

$u+\frac{1}{u}+2=0$ where $u=\frac{\tan x}{\tan 2x}$
$(u+1{)}^2=0$
$\therefore u=-1=\frac{t(1-t^2)}{2t}$
where $t=\tan x$
$\therefore -2=1-t^2$ or $t^2=3$
$\therefore t=\pm \sqrt{3}=\tan x$
$\therefore x=n\pi \pm \frac{\pi }{3}$.