To solve this equation, we can use the double angle formula for sine which states that:

sin(2x) = 2sin(x)cos(x)

Now, let's rewrite the equation with the double angle formula:

sin^2(x) - 0.5(2sin(x)cos(x)) = 0

Expand the equation:

sin^2(x) - sin(x)cos(x) = 0

Now, we can factor out sin(x) to solve for sin(x):

sin(x)(sin(x) - cos(x)) = 0

This equation will be true if either sin(x) = 0 or sin(x) - cos(x) = 0.

sin(x) = 0:
This means x = nπ, where n is an integer.

sin(x) - cos(x) = 0:
Rearrange the equation: sin(x) = cos(x)
This implies that tan(x) = 1
So, x = π/4 + nπ, where n is an integer.

Therefore, the solutions to the equation sin^2(x) - 0.5sin(2x) = 0 are x = nπ or x = π/4 + nπ, where n is an integer.