This question is essentially the same as this earlier question of mine. However, since Jacobi theta functions might be misleading in the title, in the following I just ask the exact question without prior history to it.
How to prove the following equality?
$$\prod_{n=1}^\infty(1-q^{2n-1})=\prod_{n=1}^\infty\left(\frac{1-q^n}{1-q^{2n}}\right)$$
$q=e^{2\pi i \kappa}$ for some $\kappa\in\mathbb{C}$ can be assumed.
Thanks to a hint by Qiaochu Yuan the solution becomes completely trivial. Multiplying both sides by $\prod_{n=1}^\infty\left(1-q^{2n}\right)$ we have
$$\prod_{n=1}^\infty(1-q^{2n-1})\left(1-q^{2n}\right)=\prod_{n=1}^\infty\left(1-q^{n}\right)$$
On the left hand side the first bracket probes terms with $(1-q^{\text{odd}})$ while the second bracket probes $(1-q^{\text{even}})$. Multiplying both together straightforwardly gives the right hand side.