Calculation of $\lim_{t\to+\infty}\int_{0}^{\pi} dx \cos\left(At\sin\left(\frac x2\right)\right)\cos^2\left(\frac{2n-1}2x\right)$

Question

Could someone explain me how to determine the following limit:

$\lim_{t\to+\infty}\int_{0}^{\pi} dx \cos\left(At\sin\left(\frac x2\right)\right)\cos^2\left(\frac{2n-1}2x\right)$

for $A\in \mathbb R^+$ and $n\in \mathbb N$? Is it eventually possible to reduce the integral to a Bessel function?

Thanks in advance.