# Deriving of an addition formula of $\sin$ from a second order differential equation

How to derive the addition formula of $\sin$: $$\sin(t+s)=\sin(t)\cos(s)+\sin(s)\cos(t)$$

from the following differential equation $$y''+y=0.$$

• Both sides of the first equation are solutions of $y''+y=0$ with the same initial conditions. Note that a solution of $y''+y=0$ with $y(0)=y_0$ and $y'(0)=y_1$ is unique. Then both sides are equal. – Fakemistake Jun 10 '17 at 9:39

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.