help with trig question!?

Question

tan 2x + 2sin x = 0

Answer 1

tan 2x = (sin 2x)/(cos 2x) = (2 sin x cos x)/(2 cos^2 x - 1)

(You could also use tan 2x = (2 tan x)/(1 - tan^2 x) but you would still eventually need to convert the result to sines and cosines)

Inserting this in your equation,

(2 sin x cos x)/(2 cos^2 x - 1) + 2 sin x = 0 Factoring,

2 sin x ( (cos x)/(2cos^2 x - 1) + 1) = 0.

So either sin x = 0 or (cos x)/(2 cos^2 x - 1) + 1 = 0

I will leave the rest to you.