<s>[INST] How to solve this : $ \prod^{\infty}_{n=2}(1-\frac{1}{n^2}) $ How to  solve this : 
$$ \prod^{\infty}_{n=2}(1-\frac{1}{n^2}) $$
We  can write it as follows : 
$$ \prod^{\infty}_{n=2}\frac{(n-1)(n+1)}{n^2} $$
How can we proceed further please guide.. thanks..
 [/INST] HINT:
$$\text{If }T_r=\frac{(r-1)(r+1)}{r^2},$$
$$T_{r+1}=\frac{r(r+2)}{(r+1)^2}\text{ and } T_{r-1}=\frac{(r-2)(r)}{(r-1)^2}$$
Clearly, the denominator of any $n$th term will be cancelled by the numerators of the previous & the next term for $n\ge3$
Putting $n=2,3,\cdots,n-2,n-1,n$
$$\prod_{2\le r\le n}\frac{(1\cdot3)(2\cdot4)(3\cdot5)\cdots(n-3)(n-1)(n-2)(n)(n-1)(n+1)}{(2\cdot2)(3\cdot3)(4\cdot4)\cdots(n-2)(n-2)(n-1)(n-1)\cdot n\cdot n}=\frac{n+1}{2n}$$
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