exponential function - Clever equalities proven similarly to Euler's Identity

From How to prove Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?, a very elegant proof of Euler's Identity was given. Namely, observing $f(z)=g(z)h(z)=e^{-iz}(\cos(z)+i\sin(z))$, we can see that $f'(z)=g(z)h'(z)+h(z)g'(z)=0$, showing $f(z)$ is constant, and $f(0)=1$, showing $f(z)=1$ and thus $e^{iz}=\cos(z)+i\sin(z)$.

My question is whether there are other interesting results that can be garnered by finding $g(z)$ and $h(z)$ such that $g(z)h'(z)=-h(z)g'(z)$.

Are there other interesting equalities that are known?