Evaluating closed form of $I_n=\int_0^{\pi/2} \underbrace{\cos(\cos(\dots(\cos}_{n \text{ times}}(x))\dots))~dx$ for all $n\in \mathbb{N}$.

Solution 1:

The mere fact that even the simple case $n=2$ ceases to possess a meaningful closed form in terms of elementary functions, and an entirely new function had to be invented from scratch in order to express its value, should be enough to settle all questions one might have concerning the possibility of finding such a form for larger values of the argument. Indeed, the very next case, $n=3,$ is not known to be expressible even in terms of special functions. That $\cos^{[\infty]}(x)$ is a constant $($since the function is both bound and monotone$)$ certainly constitutes a blessing, but such clearly does not hold for finite values of the iterator. $($ See also Liouville's theorem and the Risch algorithm $).$