TITLE: Show that $\cos^n{\theta}\leq\cos{n\theta},\theta\in[0,\frac{\pi}{2}],n\in]0,1[$.
QUESTION [2 upvotes]: Show that $\cos^n{\theta}\leq\cos{n\theta},\theta\in[0,\frac{\pi}{2}],n\in]0,1[$.
Can I use Taylor's polynomial?

REPLY [0 votes]: The "standard" method of solving these sorts of questions is the mean value theorem.
Let $n \in ]0,1[$ be fixed, and let $\theta \in [0, \pi / 2]$. Now consider the function
$$ f(x) = \cos n x - \cos^n x$$
Now apply the mean value theorem for the function $f$ at end points $x_1 = 0$ and $x_2 = \theta$. Once you have done this, consider why this (with a little bit more footwork) proves the statement!
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