Prove $\left(\frac{e^{\pi}+1}{e^{\pi}-1}\cdot\frac{e^{3\pi}+1}{e^{3\pi}-1}\cdot\frac{e^{5\pi}+1}{e^{5\pi}-1}\cdots\right)^8=2$

Infinite product

We proposed

(1)

$$\left(\frac{e^{\pi}+1}{e^{\pi}-1}\cdot\frac{e^{3\pi}+1}{e^{3\pi}-1}\cdot\frac{e^{5\pi}+1}{e^{5\pi}-1}\cdots\right)^8=2$$

How one go about proving (1) ?

take the ln

$$\frac{1}{8}\ln(2)=\ln\left(\frac{e^{\pi}+1}{e^{\pi}-1}\right)+\ln\left(\frac{e^{3\pi}+1}{e^{3\pi}-1}\right)+\ln\left(\frac{e^{5\pi}+1}{e^{5\pi}-1}\right)\cdots $$

May be I have to use

$$\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)=x+\frac{x^3}{3}+\frac{x^5}{5}+\frac{x^7}{7}+\cdots$$

This series is the closest I see to the product, but it seem too complicate when substitute in...

I am stuck, can't go another further please help!

Ok, this is simply Ramanujan's class invariants $g_{n}, G_{n}$ in disguise. We have \begin{align}G_{n} &= 2^{-1/4}e^{\pi\sqrt{n}/24}(1 + e^{-\pi\sqrt{n}})(1 + e^{-3\pi\sqrt{n}})(1 + e^{-5\pi\sqrt{n}})\cdots\tag{1}\\ g_{n} &= 2^{-1/4}e^{\pi\sqrt{n}/24}(1 - e^{-\pi\sqrt{n}})(1 - e^{-3\pi\sqrt{n}})(1 - e^{-5\pi\sqrt{n}})\cdots\tag{2}\end{align} Putting $n = 1$ in both the formulas and dividing them we get the expression in question as $$(G_{1}/g_{1})^{8}$$ We have $G_{1} = 1, g_{1} = 2^{-1/8}$ so the desired value $2$ is obtained easily.

It is known from the theory of elliptic integrals/functions and theta functions that $G_{n}, g_{n}$ are algebraic numbers for positive rational number $n$. Calculating these values for general rational $n$ is difficult (except for guys like Ramanujan who hand calculated a host of such values for many integers $n$).

For the current question it is sufficient to use the formulas $$G_{n} = (2kk')^{-1/12},\,g_{n} = \left(\frac{2k}{k'^{2}}\right)^{-1/12}\tag{3}$$ where $k$ is a real number with $0 < k < 1$ and $$k = k(e^{-\pi\sqrt{n}}), k' = \sqrt{1 - k^{2}}\tag{4}$$ and in general $$k = k(q) = \frac{\vartheta_{2}^{2}(q)}{\vartheta_{3}^{2}(q)}\tag{5}$$ in terms of Jacobi's Theta functions. The important thing to note here is that if $n = 1$ then $q = e^{-\pi}$ and $k = k' = 1/\sqrt{2}$ and then values of $G_{1}, g_{1}$ are easily obtained from $(3)$.