I was reading the book "Conformal Invariants" of L. Ahlfors, and seen (P.74) he says that "It is in fact obvious that the part of the teichmuller annulus (for R=1) in the upper plane is conformally equivalent to a square". Where the teichmuller annulus (for R=1) is $\mathbb{C}/(-1,0)\cup(1,\infty)$.

I don't even sure what he means, and in particular can't see why it is an obvious fact.

As I understand it, he means that there exists a conformal map from $\mathbb{H^+}/(-1,0)\cup(1,\infty)$ onto a square, **including two of it's (opposite) edges**.
without the need for the edges (only the interior) it is trivial (due to Riemann mapping theorem) and the map is well known (Schwarz–Christoffel in this case is just the elliptic integral of first kind).
But the interior is not enough since I (and Ahlfors too) need to calculate the extremal length (\conformal modulus) of the region using this fact [the interior is also equivalent to any other rectangle, while they have diffrent ext. length].
Can someone help me here please?

In fact, i am trying to find an elementary argument for the more general fact (also Page 74) that the modulus of teichmuller annulus with some specific R is (up to a multiplicative constant) the inverse of the modulus of teichmuller annulus with 1/R ($\Lambda(R)\Lambda(R^{-1})={1\over4}$). Any idea how to prove this will be usefull.