To simplify the given equation, we can rewrite sin(x/2) as 2sin(x/4)cos(x/4) using the double angle formula for sine.

So, sin(x) + [2sin(x/4)cos(x/4)]^2 = cos^2(x/2)

Expanding the square of [2sin(x/4)cos(x/4)]^2:

sin(x) + 4sin^2(x/4)cos^2(x/4) = cos^2(x/2)

Now, using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we know that sin^2(x) = 1 - cos^2(x).

Substitute sin^2(x) with 1 - cos^2(x) in the equation:

sin(x) + 4(1 - cos^2(x/4))cos^2(x/4) = cos^2(x/2)

Expand and simplify further:

sin(x) + 4cos^2(x/4) - 4cos^4(x/4) = cos^2(x/2)

Now, we can use the double-angle identity for cosine, cos(2A) = cos^2(A) - sin^2(A), to simplify the equation further:

sin(x) + 4cos^2(x/4) - 4[1 - sin^2(x/4)]^2 = [cos(x/2)]^2

Simplify the equation as much as possible, and it will be the final simplified form of the given equation.