A curious prime fraction formula
Mathematica, 32 bytes

1##&amp;@@(1-2/(Prime@Range@#^2+1))&amp;


An unnamed function that takes integer input and returns the actual fraction.

This uses the fact that (p2-1)/(p2+1) = 1-2/(p2+1). The code is then golfed thanks to the fact that Mathematica threads all basic arithmetic over lists. So we first create a list {1, 2, ..., n}, then retrieve all those primes and plug that list into the above expression. This gives us a list of all the factors. Finally, we multiply everything together by applying Times to the list, which can be golfed to 1##&amp;.

Alternatively, we can use Array for the same byte count:

1##&amp;@@(1-2/(Prime~Array~#^2+1))&amp;