Simplification of Complex number summation

Question

How does the author of this book make these simplifications from left to right? It doesn’t seem obvious. Here $h,\alpha$ are real numbers and $N$ is a natural number

$$ \left|\sum_{n=1}^{N}e^{2\pi i h n \alpha}\right|\le \frac{2}{|e^{2\pi i h \alpha}-1|}= \frac{1}{|\sin(\pi h \alpha)|} $$

Answer 1

$\sum_{n=1}^Ne^{2\pi ih n \alpha}=\frac{e^{2\pi ih (N+1) \alpha}-e^{2\pi i h \alpha}}{e^{2\pi h i \alpha}-1}$ since we have a geometric series.

Now notice that the numerator is at most $2$ in absolute value which gets you the first inequality, while the denominator absolute value is $|2\sin \pi h \alpha|$ by the double angle formula ($\cos 2y -1=-2\sin^2 y, i\sin 2y=2i\sin y \cos y, y=\pi i h \alpha$, etc), so one gets the second equality

Answer 2

Set $$ q=e^{2\pi ih\alpha} $$ then your sum is $$ \sum_{n=1}^Nq^n=\sum_{n=0}^Nq^n-1=\frac{1-q^{N+1}}{1-q}-1=q\frac{1-q^{N}}{1-q} $$ Can you conclude?