Let $f(z)=\prod_{n=1}^{\infty}(1+z^n)$ and $g(z)=\prod_{n=1}^{\infty}(1-z^{2n-1}).$
Let $\Omega=\{z\in \mathbb{C}, \; |z|<1\}.$
Since the series $\sum_{n=1}^{\infty}|z|^n$, and $\sum_{n=1}^{\infty}|z|^{2n-1}$ converges uniformly on every compact $K$ of $\Omega$ then wen can deduce the product converges and $f$ and $g$ are holomorphic on $\Omega$.
But I couldn't prove that $f(z).g(z)=1.$
$\textbf{My method:}$ I decided to determine $f$ explicitly.
Since $f$ is holomorphic and we define $f_n(z)=z^n$, then : $$\frac{f'}{f}=\sum_{n=0}^{\infty}\frac{f_n'}{f_n}=\sum_{n=0}^{\infty}\frac{n.z^{n-1}}{1+z^n}.$$ but I can't go any further