複素指数函数の指数法則と円函数の加法定理は同値

導出

$ \begin{aligned}&e^{i\theta}e^{i\phi}=e^{i(\theta+\phi)}\\\iff&(\cos\theta+i\sin\theta)(\cos\phi+i\sin\phi)=\cos(\theta+\phi)+i\sin(\theta+\phi)\\\iff&\cos\theta\cos\phi-\sin\theta\sin\phi+i(\sin\theta\cos\phi+\cos\theta\sin\phi)=\cos(\theta+\phi)+i\sin(\theta+\phi)\\\iff&\begin{dcases}\cos(\theta+\phi)=&\cos\theta\cos\phi-\sin\theta\sin\phi\\\sin(\theta+\phi)=&\sin\theta\cos\phi+\cos\theta\cos\phi\end{dcases}\end{aligned}