Assume x and y are the acute angles of a right triangle, then x + y = 90. This means that x and y are compliments. Recall that the sine of an angle is equal to the cosine of its compliment, and thus we can let cosy = sinx and cosx = siny.
Now sin(x + y) = sinxcosy + sinycosx = sinx(sinx) + siny(siny) = sin^2x + sin^2y
Now for the converse, we use the contrapositive. Assume sin(x + y) not= sin^2x + sin^2y....?