Introduction to Limits · Superchallenge problem

Let $P_n = \cos\dfrac{\pi}{4}\cdot\cos\dfrac{\pi}{8}\cdot\cos\dfrac{\pi}{16}\cdots\cos\dfrac{\pi}{2^{n+1}}$ (a product of $n$ cosines). Find $\displaystyle\lim_{n \to \infty} P_n$.