The given equation is:

cos(2x) + sin^2(x) = sin(x)

Using the trigonometric identity cos(2x) = 1 - 2sin^2(x), we can rewrite the equation as:

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

Simplifying, we get:

1 - sin^2(x) = sin(x)

Rearranging terms, we get:

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

This is a quadratic equation in sin(x), which can be solved using the quadratic formula:

sin(x) = [-1 ± √(1 + 4)] / 2
sin(x) = [-1 ± √5] / 2

Therefore, the solutions for sin(x) are:

sin(x) = (-1 + √5) / 2
sin(x) = (-1 - √5) / 2

Since sin(x) cannot be greater than 1 or less than -1, the only valid solution is:

sin(x) = (-1 + √5) / 2