How do i show that:$\prod\frac{p^2+1}{p^2-1}=\frac{5}{2}$ without using properties of Riemann zeta function?

This is a well-known problem, attributed to Sam Wagstaff in Richard Guy's Unsolved Problems in Number Theory. Section B48 "Products taken over primes" includes a paragraph

Wagstaff asked for an elementary proof (e.g., without using properties of the Riemann zeta-function that $$\prod_p \frac{p^2+1}{p^2-1} = \frac52 $$ where the product is taken over all primes. It seems very unlikely that there is a proof which doesn't involve analytical methods. At first glance it might appear that the fractions might cancel, but none of the numerators are divisible by 3 [. . .]

with a reference to

David Borwein & Jonathan M. Borwein, On an intriguing integral and some series related to $\zeta(4)$, Proc. Amer. Math. Soc., 123 (1995) 1191-1198; MR 95e:11137.

Yes, as has been noted several times in comments, this has come up before, with a beautiful answer by David Speyer: Computing $\prod_p(\frac{p^2-1}{p^2+1})$ without the zeta function?

It seems to me this should put the present question to rest. The only "property" of the Riemann zeta function used is the Euler expansion $\sum_{n \geq 1} \frac1{n^s} = \prod_p \frac1{1 - p^{-s}}$, but the proof of this boils down to the fundamental theorem of arithmetic, which goes back to Euclid's Elements I think, so I'd hardly call this using (analytic) properties of the zeta function. Aside from that, David's demonstration (which is elementary in the technical sense) does the rest.