Prove the square of the sum of two functions equals 1

Prove the square of the sum of two functions equals 1

If you have $f'(x)=g(x)$, $g'(x) = -f(x)$, $f(0)=0$ and $g(0)=1$, how do
you prove that $f^2(x)+g^2(x) = 1$?