Evaluate the infinite product $\prod_{n=1}^{\infty} \left(1-\frac{2}{(2n+1)^2}\right)$

Hint. One may recall Euler's infinite product for the cosine function

$$\cos x =\prod_{n=0}^\infty \left(1-\frac{4x^2}{(2n+1)^2\pi^2}\right),\qquad |x|<\frac \pi2.$$

Maple says this is $$\sin(\pi (\sqrt{2}-1)/2)$$ and more generally $$ \prod_{n=1}^\infty \left(1 - \frac{t^2}{(2n+1)^2} \right) = \frac{\sin(\pi (1+t)/2)}{1-t^2} $$