For an integer $n \geq 3$, define

$$A_n = \int\limits_{-\pi}^\pi\frac{|\sin(x)|}{|\cos(nx)|^{2/n}}dx.$$

It is a fact that $A_n$ is finite for all such $n$. I am interested in the behaviour of a sequence $A_n$. From the integral representation of the Beta function

$$B(x, y) = 2\int_0^{\frac{\pi}{2}}(\sin \theta)^{2x - 1}(\cos \theta)^{2y - 1}d\theta,$$

it is rather easy to derive an upper bound

$$A_n \leq 2B\left(\frac{1}{2}, \frac{1}{2} - \frac{1}{n}\right).$$

However, I think that more accurate estimates can be made. In particular, note that $2B\left(\frac{1}{2},\frac{1}{2} - \frac{1}{n}\right)$ approaches $2\pi$ as $n$ tends to infinity, whereas I believe that $A_n$ should approach a value strictly smaller. What is the limit? Can we say that $A_n$ is monotonously decreasing? I would value any suggestions on how to approach this problem.

Below is the graph of the function $\left|\sin(x)\cos(nx)^{-2/n}\right|$ for $x \in [-\pi, \pi]$ for $n = 17$.