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.

$$\frac{1}{1-e^{i\pi/7}}=\frac{e^{-i\pi/14}}{e^{-i\pi/14}-e^{i\pi/14}}=\frac12\frac{ie^{-i\pi/14}}{\sin\frac\pi{14}}=\frac12+\frac i2\frac{\cos\frac{\pi}{14}}{\sin\frac{\pi}{14}}=\frac{1}{2}+\frac i2\cot\frac{\pi}{14}$$
