Equality between an expression with $\sin x$ and $e^{it}$

Question

I define Dirichlet's kernel in the following way: $$D_N(t) = \sum \limits_{k = -N}^{N} e^{2 \pi i kt}.$$ I managed to show that: $$D_N(t) = \sum \limits_{k = -N}^{N} e^{2 \pi i kt} = e^{-2 \pi i Nt} \sum \limits_{k = 0}^{2N} e^{2 \pi i kt} = e^{-2 \pi i Nt} \frac{e^{2 \pi i (2N+1)t} -1}{e^{2 \pi it} -1}.$$ My task was however to show that: $$D_N(t) = \frac{\sin \big(\pi(2N + 1)t \big)}{\sin(\pi t)}.$$ The equality must be quite trivial but I've spent some pages with no result. I would appreciate any hints or tips.

Answer 1

You are almost done. Just multiply numerator and denominator additionally by $e^{-\pi i t} $ and use $e^{i x} -e^{-i x}=2i\sin x $.