Evaluate $\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$ Difficult question from some test somewhere (I forget).

$$\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$$

$x$ is, of course, an integer.

Write 
$$\frac{x^4 - 1}{x^4+1} =  \frac{(x-a_1)\ldots(x-a_4)}{(x-b_1)\ldots(x-b_4)}$$
where $a_j$ are the roots of $x^4-1$ and $b_j$ are the roots of $x^4+1$.
Then the partial product
$$ \prod_{x=2}^n \frac{x^4 - 1}{x^4+1} = \frac{\Gamma(n+1-a_1)\ldots \Gamma(n+1-a_4) \Gamma(2-b_1) \ldots \Gamma(2-b_n)}{\Gamma(2-a_1) \ldots \Gamma(2-a_4) \Gamma(n+1-b_1) \ldots \Gamma(n+1-b_4)}$$
Now (carefully) take the limit as $n \to \infty$.
