A recent topic on MSE involved dealing with the sequence $(x_n)$ defined by $0 < x_0 < \pi$ and $$x_{n+1}=\int_0^{x_n} \cos^n(t) \, dt$$ for all integers $n \geq 0$. I am wondering how to find an equivalence of $x_n$ when $n \to +\infty.$
Many thanks.