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Harvard-MIT November Tournament

Algebra Difficulty 4.8 AIME Prove it United States

Problem:
What is the period of the function $f(x) = \cos(\cos(x))$ ?

Solution

Solution:
$\boxed{\pi}$

Since $f(x)$ never equals $\cos(1)$ for $x \in (0, \pi)$ but $f(0) = \cos(1)$ , the period is at least $\pi$ . However, $\cos(x+\pi) = -\cos(x)$ , so $\cos(\cos(x+\pi)) = \cos(\cos(x))$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.