I have to find the maximum of $|\cos(z)|$ on a circle given bij $|z| = \pi k$ ($k \in \mathbb{Z}_{>0})$. I found that you could use the Taylor series expansion of $\cos(z)$ to find the following:
$$|\cos(x)| \leq 1 - \frac{|x^2|}{2!} + \frac{|x^4|}{4!} - \dots$$
This is maximal for $x = i$ if we only look at the unit circle. But how can I expand this to a "random" circle given by $|z| = \pi k$?
The correct estimate is $$ |\cos(z)| = \left| 1 - \frac{z^2}{2!} + \frac{z^4}{4!}+ \ldots\right| \le 1 + \frac{|z|^2}{2!} + \frac{|z|^4}{4!} + \ldots = \cosh(|z|) \, . $$ Equality holds for $z = \pm iy$, $y \in \Bbb R$. It follows that for any radius $r \ge 0$ $$ \max_{|z|=r} |\cos(z)| = \cosh(r) \, . $$