algebra precalculus - The number of solutions of the equation $|cot x|=cot x +frac{1}{sin x}$ where $0

$$|\tan x| = \frac{\sin x}{1+\cos x}$$
$$|\tan x|=\tan (\frac x2)$$ then x can assume only two values $0,2\pi$ (at least that’s what I think). But since it doesn’t fit in the given interval, number of solutions should be zero, but the answer is 2. How should I correct it?