My question is whether someone could provide a proof for the following identity:
$$ \frac{1 - e^{int}}{1 - e^{it}} = e^{i(n-1)t/2} \frac{\sin(nt/2)}{\sin(t/2)} $$
Motivation:
The left hand side is the sum of the geometric series of complex exponentials, which is itself the fourier transform of a finite Dirac Comb. This sum appears in the derivation of the Frauenhofer Diffraction pattern in Physics.
$$ \mathcal{F}[\sum_{n=0}^{n=N-1} \delta(t - n)] \sum_{n=0}^{n=N-1}e^{int} = \frac{1 - e^{int}}{1 - e^{it}} $$