あまり見る機会がないですが，以下です．$j$は虚数です．
$$e^{-j\alpha}-e^{-j\beta}=-2j\mbox{sin}(\displaystyle\frac{\alpha-\beta}{2})\cdot e^{-j\frac{\alpha+\beta}{2}}$$
$$e^{-j\alpha}+e^{-j\beta}=2\mbox{cos}(\displaystyle\frac{\beta-\alpha}{2})\cdot e^{-j\frac{\alpha+\beta}{2}}$$
$$e^{j\alpha}-e^{j\beta}=-2j\mbox{sin}(\displaystyle\frac{\beta-\alpha}{2})\cdot e^{j\frac{\alpha+\beta}{2}}$$
$$e^{j\alpha}+e^{j\beta}=2\mbox{cos}(\displaystyle\frac{\alpha-\beta}{2})\cdot e^{j\frac{\alpha+\beta}{2}}$$