I'm working on a homework problem, and a part of it is to show that $|\cos z| \leq e$ if $|z|=1$, where $z$ is a complex number.

I figured that it would be useful to write $\cos(z) = \frac{e^{iz}+e^{-iz}}{2}$, and then by the triangle inequality, we can say that $|\cos(z)| \leq 1/2 |e^{iz}| + 1/2 |e^{-iz}|$, however working out the two norms yields $1/2(2)=1$, which isn't very useful.

Any hints would be greatly appreciated.