in the interval 0 'less than or equal to' x 'less that or equal to' pi

cos 2x - sin x = 0      note:  cos 2x =  1 - 2sin^2x

1 - 2sin^2x - sinx    =   0      multiply through by -1

2sin^2x  + sin x   - 1   = 0     factor

(2sinx - 1) (sinx + 1)  =  0      set each factor to 0

2sinx  - 1  = 0    add 1 to both sides

2sinx  = 1      divide both sides by  2

sinx   = 1/2       and this happens at   pi/6  and 5pi/6   on the requested interval

And

sin x + 1 = 0    subtract 1 from both sides

sin x = -1       this happens at 3pi/2, which is outside the requested interval