Why is $\left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24} = \frac{1}{q \, \Pi_{n\geq1}(1+q^n)^{24}}$

Question

Where the Dedekind eta function,

$$\eta(\tau) = q^{1/24} \prod_{n=1}^\infty (1-q^n)$$

and $q = \exp(2\pi i\tau)$.

I cant seem to get the equality, am I missing some identities?

Thanks!

Answer 1

$\eta(\tau)^{24}=q\prod_{n=1}^{\infty}(1-q^n)^{24}$ and $\eta(2\tau)^{24}=q^2\prod_{n=1}^{\infty}(1-q^{2n})^{24}$, so by writing $$(1-q^{2n})=(1-q^n)(1+q^n)$$ we get $$ \Big(\frac{\eta(\tau)}{\eta(2\tau)}\Big)^{24}=\frac{q\prod_{n=1}^{\infty}(1-q^n)^{24}}{q^2\prod_{n=1}^{\infty}[(1-q^n)(1+q^n)]^{24}}=\frac{1}{q\prod_{n=1}^{\infty}(1+q^n)^{24}}$$