Jacobi's two-square theorem and related identities (Q1306581)

It is shown that Jacobi's two-square theorem is an almost immediate consequence of a famous identity of Jacobi \[ \prod_{n=1}^\infty (1-x^n)^3= \sum_{m=0}^\infty (-1)^m (2m+1) x^{\frac 12 m(m+1)}. \] Furthermore, the author draws combinatorial conclusions from two identities of Ramanujan, namely a formula for the number of representations of an integer as a sum of three squares resp. three triangular numbers.