Determine the exact value of x.

 cos2x -  sinx = 0

(1 - 2sin^2 x) - sin x  = 0             multiply through by -1

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

(2sin x - 1) (sin x + 1) = 0

Set each factor to 0   and solve for  x

2sinx - 1 = 0

2sin x = 1

sin x = 1/2

And this happens at  x =   pi/6 + 2pi*n       and     x =   5pi/6 + 2pi*n       where n is an integer

Also

sinx + 1  = 0

sin x = -1

And this happens at     x  =   3pi/2 + 2pi*n    where n is an integer