In their recent paper, Farkas and Kra proved five three-term identities for the coefficient of certain finite products. In this paper we will show that each of the five identities is just one special case of an infinite family of identities. We will briefly introduce modular forms, and show that the coefficients of these products are closely related to the coefficients of certain modular form. Consequently in our proof we will use the properties of modular forms, specifically the Hecke operators.