How to derive the addition formula of $\sin$: $$ \sin(t+s)=\sin(t)\cos(s)+\sin(s)\cos(t) $$
from the following differential equation $$ y''+y=0. $$
Define $\phi(t)=\sin(t+s)$ and $\psi(t)=\sin t\cos s+\sin s\cos t$ and show that they are solutions of the given differential equation. Show then $\phi(0)=\psi(0)$ and $\phi'(0)=\psi'(0)$ and argue with uniqueness.