Mobius transform and defining an hyperbolic angle

Solution 1:

The inner product of two complex numbers (identifying $\mathbb{C}\cong\mathbb{R}^2$) is $\langle w,z\rangle=\mathrm{Re}(\overline{w}z)$.

The chain rule says $(A\circ\alpha)'(0)=A'(\alpha(0))\alpha'(0)$ and similarly for $(A\circ\beta)'(0)$.

You should be able to start simplifying the right side now and show you end up with the same thing as the left side. Notice this only depends on $A$ being holomorphic in a neighborhood of $\zeta$, so this fact about preserving angles (conformality) is much more general than Mobius transformations.

Note that $\lambda\overline{\lambda}=|\lambda|^2$ and real scalars can be pulled out of $\mathrm{Re}(\,)$.