How do you find the general solutions for sinx+2tanx=0?

Note that since $\tan x = \sin \frac{x}{\cos} x$, $\sin x = \cos x \tan x$
$\sin x + 2 \tan x = 0$
$\cos x \tan x + 2 \tan x = 0$
$\tan x \left(\cos x + 2\right) = 0$
$\cos x = - 2$ (no solutions) or $\tan x = 0 \setminus R i g h t a r r o w x = k \pi , k \in \mathbb{Z}$