Show that $f^2 - g^2 = C$

Question

$f$ and $g$ are continuously differentiable functions such that $f(x) = g'(x)$ and $g(x) = f'(x)$ and any product of $f, g, f', g'$ is commutative for all $x \in \mathbb R$

I need to show that $f^2 - g^2 = C$ , where $C$ is a constant.

Logically this makes sense if you for example use a trig function $\sinh(x)$ but how can I show this to be true?

Answer 1

Note that$$(f^2-g^2)'=2ff'-2gg'=2fg-2gf=0.$$