To solve this trigonometric equation, we can first simplify it by using the trigonometric identities:

sin^2(π - x) + cos(π/2 + x) = 0
sin^2(π - x) + sin(π/2 - x) = 0

Now, we can use the trigonometric identity sin(π - x) = sin x and cos(π/2 - x) = sin x to simplify the equation further:

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

Now, we have two possibilities for sin x:

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

1) If sin x = 0, then x = 0, π

2) If sin x + 1 = 0, then sin x = -1, which has no real solutions.

Therefore, the solutions to the equation sin^2(π - x) + cos(π/2 + x) = 0 are x = 0, π.