Verifying that $ \prod_{j=1}^{\infty} \frac{1}{1-q^j} = \prod_{j=1}^{\infty} \frac{1}{(1-q^{2j-1})(1-q^{2j})}$ On page 165 of Chapter 13, how was the equality made from line 1 to line 2?
https://archive.org/details/NumberTheory_862/page/n173
Namely, how $$ \prod_{j=1}^{\infty} \frac{1}{1-q^j} = \prod_{j=1}^{\infty} \frac{1}{(1-q^{2j-1})(1-q^{2j})}$$

Simply pair-off factors and use the fact that multiplication is commutative:
\begin{align*}
\prod_{j=1}^{\infty}\frac{1}{(1-q^{j})} & = \frac{1}{1-q}\frac{1}{1-q^2}\frac{1}{1-q^3}\frac{1}{1-q^4}\cdots\\
& = \left(\frac{1}{1-q}\frac{1}{1-q^2}\right)\left(\frac{1}{1-q^3}\frac{1}{1-q^4}\right)\cdots\\
& = \left(\frac{1}{1-q^{2}}\frac{1}{1-q}\right)\left(\frac{1}{1-q^4}\frac{1}{1-q^{3}}\right)\cdots\\
& = \prod_{j=1}^{\infty}\frac{1}{(1-q^{2j})(1-q^{2j-1})}.
\end{align*}
