Neat way to prove $\sin(\alpha+\beta)$ using complex exponential

Question

I am supposed to prove that $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha$ using complex exponentials:

$$ \begin{align} \sin\theta&=-\frac{1}{2}i(e^{i\theta}-e^{-i\theta})\\ \cos\theta&=\frac{1}{2}(e^{i\theta}+e^{-i\theta}) \end{align} $$

The proof that I have done is very long and messy and essentially I am showing that LHS and RHS are the same thing. I was wondering if there is a neater proof?

Answer 1

Hint

Take the imaginary part of $$e^{i(\alpha+\beta)}=e^{i\alpha}e^{i\beta}$$