Find the Product: the answer is a product

We know:

$\frac {\sin{x}}{x}= \left(1-\frac {x^2}{\pi^2}\right)\left(1-\frac {x^2}{4\pi^2}\right)\left(1-\frac {x^2}{9\pi^2}\right)\left(1-\frac {x^2}{16\pi^2}\right)\cdots$

Putting $x=\pi/10$ and knowing that $\sin(\pi/10)=\displaystyle \frac 1{2\phi}$ where, $\displaystyle \phi=\frac {1+\sqrt{5}}{2}$ we get,

$\frac {5}{\pi\cdot\phi}=\left(1-\frac 1{{10}^2}\right)\left(1-\frac 1{{20}^2}\right)\left(1-\frac 1{{30}^2}\right)\cdots$

hence,

$\frac {100\cdot400\cdot900\cdot1600\cdot2500\cdots}{99\cdot399\cdot899\cdot1599\cdot2499\cdots} = \frac {\pi\cdot\phi}{5}$

$\therefore A \approx \boxed{5.0832}$