Find all the entire functions satisfying $f^2+g^2=1$

Question

Find all the entire functions (holomorphic functions on $\mathbb{C}$) $f,g$ satisfying $f^2+g^2=1$.

Of course $f,g$ can be constants satisfying $f^2+g^2=1$. But if they are not constants, do they exist?

Maybe Liouville Theorem can be used here to solve this problem, but how to use?

Answer 1

Since $f^2+g^2=1$, $(f+ig)(f-ig)=1$. So, $f+ig$ is an entire function without zeros, and therefore it can be written as $e^{hi}$, for some entire function $h$. So,$$f-ig=\frac1{f+ig}=\frac1{e^{hi}}=e^{-hi}.$$Therefore,$$f=\frac{f+ig+f-ig}2=\frac{e^{hi}+e^{-hi}}2=\cos\circ h$$and$$g=\frac{f+ig-(f-ig)}{2i}=\frac{e^{hi}-e^{-hi}}{2i}=\sin\circ h.$$