Let $\lambda(\tau)=\dfrac{e_3(\tau)-e_2(\tau)}{e_1(\tau)-e_2(\tau)}$, where $e_1=\wp(\frac{1}{2})$, $e_2=\wp(\frac{\tau}{2})$, and $e_3=\wp(\frac{1+\tau}{2})$. (Please see this question for the complete set-up.)
There are $5$ functional equations satisfied by $\lambda$, three of them being consequences of the "basic" two, namely $\lambda(\tau+1)=\frac{\lambda(\tau)}{\lambda(\tau)-1}$ and $\lambda(-1/\tau)=1-\lambda(\tau)$.
In a couse which I followed some time ago those two were proved using some specific results and notation from earlier lectures. (The main result I mean is that there exists a holomorphic automorphism of $\mathbb C$ mapping $L(\tau_1)$ to $L(\tau_2)$ iff there exists a biholomorphism between the complex tori corresponding to those two lattices; and the proof of the functional equations was based heavily on explicit forms of those maps and notation intoduced along the way.) I was wondering whether there is a kind of independent proof of these relations?