To simplify the expression, let's expand the terms inside the parentheses first:

(cosx/2 - sinx/2)(cosx/2 + sinx/2)
= cos^2(x)/2 + sin^2(x)/2 - sinxcosx/2 - sinxcosx/2
= cos^2(x)/2 + sin^2(x)/2 - sinxcosx

Now, substituting this expression back into the original equation:

sinx + (cosx/2 - sinx/2)(cosx/2 + sinx/2) = 0
sinx + (cos^2(x)/2 + sin^2(x)/2 - sinxcosx) = 0
sinx + (1/2) = 0
sinx = -1/2

So, the solution to the equation sinx + (cosx/2 - sinx/2)(cosx/2 + sinx/2) = 0 is sinx = -1/2.