Using Jacobi's Triple Product Identity prove that $$\begin{aligned} \prod_{n=1}^{\infty}\left(1-x^{n}\right)^{6}=& \sum_{n=-\infty}^{\infty} x^{n^{2}} \sum_{n=1}^{\infty}(2 n+1)^{2} x^{n(n+1)} \\ &-\sum_{n=-\infty}^{\infty} x^{n(n+1)} \sum_{n=1}^{\infty}(2 n)^{2} x^{n^{2}} \end{aligned}$$
I found this identity in the article COMPLETION OF A GAUSSIAN DERIVATION - JOHN A. EWELL See the image below
He wrote the proof for $|x|<1$. But if you see in the article A SIMPLE PROOF OF JACOBI'S FOUR-SQUARE THEOREM - M. D. HIRSCHHORN he used the identity without such constriants. See the image below
Then what will be the generalised proof of this identity. Also please make the proof simple. I have seen many proofs of Jacobi's Four Square Theorem but only this Sumple proof i understood as i dont know much that level mathematics