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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?

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$\cos(x+iy) = \cos x \cosh y - i \sin x \sinh y$ so $$|\cos(x+iy)|^2 = \cos^2 x \cosh^2 y + \sin^2 x \sinh^2 y = \dfrac{\cos(2x)}{2} + \dfrac{\cosh(2y)}{2}$$

On the lines $x = \pm n \pi \pm y$, $\cos(2x) = \cos(2y)$ so $$|\cos(x+iy)|^2 = \frac{\cos(2y)+\cosh(2y)}{2} = \sum_{n=0}^\infty \frac{(2y)^{4n}}{(4n)!} \ge 1$$ Now $\cosh(2y)$ is an increasing function of $|y|$, and the square with vertices $[\pm n\pi, 0]$ and $[0,\pm n\pi]$ is inscribed in the circle $|x+iy|=n\pi$, so $|\cos(x+iy)|^2 \ge 1$ on that circle.

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Fantastic, Robert! What a clever strategy. Thanks! –  user49678 Nov 16 '12 at 15:31
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