Find simple proofs of the two $q$-series identities

Question

When I read an article, I found the following two $q$-series identities very interesting $$ \sum_{k=-\infty}^{+\infty}(-1)^k{2n\brack n+2k}q^{2k^2}=(-q;q^2)_n, $$ $$ \sum_{k=-\infty}^{+\infty}(-1)^k{2n\brack n+2k}q^{3k^2+k}=(-q;q)_n, $$ where $(a;q)_n=(1-a)(1-aq)\cdots (1-aq^{n-1})$ and $${n\brack m}=\frac{(q;q)_n}{(q;q)_m(q;q)_{n-m}}.$$ Letting $n\to \infty$ in the above two identities, we obtain Gauss's identity for square numbers and Euler's pentagonal number thorem. The proofs of this two identities are complicated. Can anyone give simple proofs of them?