Given the sequence defined by $$u_0=1\quad \text{and}\quad u_{n+1}=\frac{u_n}{|\cos(u_n)|^{\alpha}}$$
Numerically, it appears that for $\alpha=1$, $u_n^{\frac{1}{n}}$ tends to $2$, and for $\alpha=\frac{1}{2}$, $u_n$ tends to $\pi$. However, I have no idea how to prove it. And it seems very difficult to find all $\alpha>0$ for which the sequence converges.