Let $n$ be a positive integer.
Prove that $|\cos z|\geq 1$ for all $z\in\mathbb{C}$ lying on the circle of radius $n\pi$, centered at the origin.
In other words, prove that $|\cos(n\pi e^{i\theta})|\geq 1$ for all real $\theta$ and $n=1,2,3,\dots$.
What is the most elegant way to prove this?