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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?

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Hint

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

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  • $\begingroup$ Can't get more succinct than that! You are great at saying what needs to be said very succinctly! $\endgroup$ – Namaste Mar 16 '14 at 13:15
  • $\begingroup$ Thanks dear amWhy but If I improve my English I would give longer answers;-) $\endgroup$ – user63181 Mar 16 '14 at 13:18
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    $\begingroup$ You actually have a pretty "darn" good mastery of English, Sami! Seriously. $\endgroup$ – Namaste Mar 16 '14 at 13:20

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