Find real part of $\frac{1}{1-e^{i\pi/7}}$ How can you find
$$\operatorname{Re}\left(\frac{1}{1-e^{i\pi/7}}\right).$$
I put it into wolframalpha and got $\frac{1}{2}$, but I have no idea where to begin. I though maybe we could use the fact that $$\frac{1}{z}=\frac{\bar{z}}{|z|^2},$$ where $\bar{z}$ is the conjugate of $z$. Unfortunately, the magnitude doesn't seem to be a nice number. I feel like this might be a trigonometry question in disguise, but converting $e^{i\pi/7}=\cos\left(\frac{\pi}{7}\right)+i\sin\left(\frac{\pi}{7}\right)$ hasn't been very fruitful.

Let
$$z=\frac1{1-e^{it}}$$
where $t$ is real and $e^{it}\ne1$.
Then
$$z+\overline z=\frac1{1-e^{it}}+\frac1{1-e^{-it}}
=\frac1{1-e^{it}}+\frac{e^{it}}{e^{it}-1}=
\frac{1-e^{it}}{1-e^{it}}=1.$$
Therefore the real part of $z$ equals $1/2$.
