A post in physics stack exchange led me to get confused in one of the steps where the summation of a Euler form suddenly became a trigonometric expression. The answer to the post I got confused in was:
In the answer (there is only one answer to the linked post), there is a summation which I do not get the result it yields. I do get one should use geometric series to approach it, and I can get the numerator expression (due to there being $2N+1$ terms), but I am having trouble seeing the denominator expression.
$$\sum_{n=-N}^Ne^{iknd\sin \theta}=\frac{\sin\big((N+\frac{1}{2})kd\sin\theta\big)}{\sin\big(\frac{1}{2}kd\sin\theta\big)}$$
Could anyone help me show how the above is proved? Thank you
Edit: attached is my working out but I do not know what to do with $e^{i\phi}$ term.
