(I don't know if you can prove it this way.)
Draw a unit circle. Plot a point P(x, y) on the unit circle. Drop a perpendicular from point P to the x-axis. You formed a right triangle. Then
According to the Pythagorean theorem,
so plug in for x and for y and you get
EDIT: Wow, I completely misread the problem. Didn't see the 2 in front of the x.