On a Curious Identity of Ramanujan

Ramanujan wrote the following identity {align*} √2 (1 - 1/3^2) (1 - 1/7^2) (1 - 1/11^2) (1 - 1/19^2) \ = \ (1 + 1/7) (1 + 1/11) (1 + 1/19). {align*} We find necessary and sufficient conditions for the integers in the identity and prove that there are only finitely many such identities, and provide a method to generate many interesting variations.