Question 184
Let $z = \cos \dfrac{2\pi}{n} + i \sin \dfrac{2\pi}{n}$ where $n$ is a positive odd integer. Prove that
$$
\frac{1}{1+z} + \frac{1}{1+z^{2}} + \frac{1}{1+z^{3}} + \cdots + \frac{1}{1+z^{n}} = \frac{n}{2}
$$
$$
\frac{1}{1+z} + \frac{1}{1+z^{2}} + \frac{1}{1+z^{3}} + \cdots + \frac{1}{1+z^{n}} = \frac{n}{2}
$$