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Here I will illustrate how to use the complex exponential to derive some useful trigonometric identities. Lets start with the Euler identities:

\begin{equation}e^{i\theta}=\cos\theta + i\sin\theta\end{equation}
\begin{equation}e^{-i\theta}=\cos\theta – i\sin\theta\end{equation}

Now put $\theta = A+B$ in equation (1) and use the rule for multiplication of exponentials:

\begin{equation}e^{i(A+B)}=e^{iA}e^{iB}\end{equation}