Recently , I read one paper titled 'Modular equations and approximations to π' by Ramanujan, in which there are some formulas for $q=\pi i \tau$( where $\tau=x+yi, y>0$, hence $|q|<1)$ :
$$\prod_{n=1}^\infty\left(1+q^{2n-1}\right)=2^{\frac{1}{6}} q^{\frac{1}{24}}(kk')^{-\frac{1}{12}} ~~~ (1)$$
and $$ \prod_{n=1}^\infty\left(1-q^{2n-1}\right)= 2^{\frac{1}{6}} q^{\frac{1}{24}}k^{-\frac{1}{12}}k'^{\frac{1}{6}} ~~~~(2)$$
where $k=k(\tau)$ is the Jacobi modulus， $k^2(\tau)=\lambda(\tau)$, the elliptic modular function, and $k'=\sqrt{1-k^2}.$
The following result can be calculated by Mathematica: $$\left(1+e^{-\pi }\right)\left(1+e^{-3 \pi }\right)\left(1+e^{-5 \pi }\right) \cdots=2^{\frac{1}{4}} e^{-\pi / 24}.$$
But I do not know how to prove these formulas (1) and (2). I would appreciate if someone could give some suggestions.
Here is a brief outline of how these identities can be proved. The material is borrowed from my blog (linked in comments to question).
Let's start with a real number $k\in(0,1)$ called modulus and define a complementary modulus $k'=\sqrt{1-k^2}$ and the complete elliptic integral of first kind: $$K(k)=\int_{0}^{\pi/2}\frac{dx}{\sqrt{1-k^2\sin^2x}}\tag{1}$$ The numbers $K(k), K(k') $ are usually denoted by $K, K'$. The real surprise here is that given the values of $K,K'$ (they are not independent but are rather functions of $k$) it is possible to find the values of $k, k'$ via the magical formulas of Jacobi: $$k=\frac{\vartheta_{2}^2(q)}{\vartheta_{3}^2(q)}, k'=\frac{\vartheta_{4}^2(q)}{\vartheta_{3}^2(q)}\tag{2}$$ where $q=e^{-\pi K'/K} $ is the nome corresponding to modulus $k$ and \begin{align} \vartheta_{2}(q)&=\sum_{n=-\infty} ^{\infty} q^{(n+(1/2))^2}\notag\\ \vartheta_{3}(q)&=\sum_{n=-\infty} ^{\infty} q^{n^2}\notag\\ \vartheta_{4}(q)&=\vartheta_{3}(-q)\notag \end{align} are theta functions of Jacobi. The formulas $(2)$ can be proved using properties of elliptic integrals and series definitions of theta functions provided above. This has been done in this and the following post.
Another ingredient needed here is the Jacobi triple product which allows us to obtain product representations for these theta functions. It can be stated as $$\sum_{n\in\mathbb {Z}} z^nq^{n^2}=\prod_{n=1}^{\infty} (1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})\tag{3}$$ for all complex numbers $z, q$ with $z\neq 0$ and $|q|<1$.
You can easily prove your identities in question via the use of $(2)$ and $(3)$.