Chapter 4.8, Problem 35ES

(a) Show that if $f$ and $g$ are functions for which $f^{\prime }\left(x\right)=g\left(x\right)$ and $g^{\prime }\left(x\right)=-f\left(x\right)$ for all $x,$ then $f^2\left(x\right)+g^2\left(x\right)$ is a constant.

(b) Give an example of functions $f$ and $g$ with this property.