limits - $lim_{n to infty} 2^ncosleft(frac{pi}{2^n}right)sinleft(frac{pi}{2^n}right)$ (Without L'Hospital)

I'm trying to find
$$\lim_{n \to \infty} 2^n\cos\left(\frac{\pi}{2^n}\right)\sin\left(\frac{\pi}{2^n}\right)$$
I think the answer is $\pi$, but I don't know how to find it.
Could you please show me the shortcut?