Here is the question:

Suppose $n$ is a positive integer, and $z$ is a complex number with modulus $1$, such that $z^{2n}$ is not $−1$, show that $\dfrac{z^n}{1+z^{2n}}$ is a real number.

A complex number with modulus $1$ takes the form $\cos\theta + i\sin\theta$.  Then the rest consists of applying identities involving complex numbers and trigonometry.