$J_n = \int_0^{\pi/2} \cos^{2n}\theta\,d\theta$

Expand $\cos^{2n}\theta$ as a trigonometric polynomial in $e^{i\theta}$ and $e^{-i\theta}$, using the binomial expansion. The constant term is

$2^{-2n} {2n\choose n}$

and the other terms are multiples of $\cos (2k\theta)$, for $k=1,2, \ldots n$
Therefore

$J_n = \int_0^{\pi/2} 2^{-2n} {2n\choose n}\,d\theta = \frac{\pi}{2} 2^{-2n} {2n\choose n}$

We are done if we can prove that $\sqrt{n} J_n \to \frac{1}{2}\sqrt{\pi}$ as $n\to\infty$