Show fnfinite products are equal

I have been asked to show that for $\displaystyle |z|<1$ that

$\displaystyle \displaystyle \left[ \prod_{j=0}^{\infty}(1-z^{2j+1})\right]^{-1}=\prod_{j=0}^{\infty}(1+z^{j+1})$

I know since $\displaystyle \sum_{j=0}^{\infty}|z^{j+1}|$ and $\displaystyle \sum_{j=0}^{\infty}|z^{2j+1}|$

Converge uniformly on compact subsets of the unit disk that both of the infinite products converge to a holomorphic function of the open disk.

So here is my idea let

$\displaystyle f= \prod_{j=0}^{\infty}(1-z^{2j+1})$ and

$\displaystyle g=\prod_{j=0}^{\infty}(1+z^{j+1})$

I would be sufficient to show that $\displaystyle fg=1$

My idea was to take the deriative and show that it is equal to $\displaystyle 0$

let $\displaystyle h=fg \implies \ln(h)=\ln(fg)$

This gives

$\displaystyle \displaystyle \frac{h'(z)}{h(z)}=\frac{d}{dz} \left( \sum_{j=0}^{\infty}\ln(1+z^{j+1})+\ln(1-z^{2j+1})\right)=\sum_{j=0}^{\infty}\frac{(j+1)z^j }{1+z^{j+1}}-\frac{(2j+1)z^{2j}}{1-z^{2j+1}}$

I cannot sum this series or show that it is equal to 0. Also if you see a different approach please let me know. (Clapping)

Thanks