TITLE: Showing that $\frac{z^n}{1+z^{2n}}$ is real for $|z|=1$.
QUESTION [5 upvotes]: Can you help me to prove that $$\frac{z^n}{1+z^{2n}}$$
is a real number, given that $z$ is a complex number with modulus $1$ and $n$ a positive integer, such that $z^{2n}$ is not equal to $-1$.

REPLY [2 votes]: $$z=a+bi\\\frac{z^n}{1+z^{2n}}=\frac{z^n\cdot \overline{z}^n}{\overline{z}^n+z^{2n}\overline{z}^n}=\frac{1}{z^n+\overline{z}^n}=\frac{1}{(a+bi)^n+(a-bi)^n}$$
By the binomial theorem all the terms with $i$'s cancel out