\begin{multline} \shoveleft \begin{aligned} e^{i\alpha} e^{i\beta} = e^{i(\alpha+\beta)} = \cos(\alpha+\beta) + i\sin(\alpha+\beta) \end{aligned} \end{multline} \begin{multline} \shoveleft \begin{aligned} e^{i\alpha} e^{i\beta} & = (\cos\alpha + i\sin\alpha)(\cos\beta + i\sin\beta) & \\ & = \cos\alpha\cos\beta - \sin\alpha\sin\beta + i(\sin\alpha\cos\beta + \sin\beta\cos\alpha) \end{aligned} \end{multline} \begin{multline} \shoveleft \begin{aligned} \cos(\alpha+\beta) &= \cos\alpha\cos\beta - \sin\alpha\sin\beta & \\ \sin(\alpha+\beta) &= \sin\alpha\cos\beta + \sin\beta\cos\alpha \end{aligned} \end{multline}