Non-Elementary Integration · Superchallenge problem

Using the Wallis product $\displaystyle\prod_{n\ge1}\frac{4n^2}{4n^2 - 1} = \frac\pi2$, compute $\displaystyle\prod_{n=1}^{\infty}\left(1 - \frac{1}{4n^2}\right)$. Express your answer in terms of $\pi$.