Solution 6. By the product formula for sine we have \[ \sin (\pi / 2)=\frac{\pi}{2} \prod_{n=1}^{\infty}\left(1-\frac{1}{4 n^2}\right)=\frac{\pi}{2} \prod_{n=1}^{\infty}\left(\frac{(2 n+1)(2 n-1)}{(2 n)(2 n)}\right) \]