New proofs for two infinite product identities (Q1209546)

The identities in question are \[ \begin{aligned} \Pi_ 1(1-x^ n)\;\Pi_ 2(1+x^ n)&=\sum_{n=0}^ \infty (x^{2n(n+1)}+x^{6n(n+1)+1})\qquad\text{ and}\\ \Pi_ 3(1-x^ n)\;\Pi_ 4(1+x^ n)&=\sum_{n=0}^ \infty (x^{n(n+1)}+x^{5n(n+1)+1}),\end{aligned} \] where the products are extended over all positive integers in the following residue classes: \(\Pi_ 1\): \(n\equiv 0,\pm5,\pm7\pmod{24}\); \(\Pi_ 2\): \(n\equiv\pm1,\pm4,6\pmod{24}\); \(\Pi_ 3\): \(n\equiv 0,\pm3\pmod{10}\); \(\Pi_ 4\): \(n\equiv\pm1,\pm2,\pm4\pmod{10}\).