sin2x + 2 cos x = 0

2sinxcosx + 2cosx = 0 divide through by 2

sinxcosx + cosx = 0 factor

cosx (sinx + 1) = 0

So either

cosx = 0 and this happens at pi/2 + pi*n where n is an integer

Or

sin x + 1 = 0

sin x = - 1 and this happens at 3pi /2 + 2pi*n where n is an integer