general solution for sin2x + 2 cos x = 0

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