an isomorphism from $L^\infty(\mathbb{T})$ to $L^\infty([-1,1],\frac{2}{\pi } \sqrt{1-t^2}\mathrm{d}t)$

Question

$\mathbb{T}$ is the boundary of unit ball.Consier $\phi:[-1,1]\rightarrow\mathbb{T},\phi(t)=e^{2i(\arcsin t+t\sqrt{1-t^2})},t\in[-1,1]$. It is easy to check that $L^2(\mathbb{T})\ni f\mapsto f\circ\phi\in L^2([-1,1],\displaystyle\frac{2}{\pi } \sqrt{1-t^2}\mathrm{d}t)$ is a unitary which implements an isomorphism from $L^\infty(\mathbb{T})$ to $L^\infty([-1,1],\displaystyle\frac{2}{\pi } \sqrt{1-t^2}\mathrm{d}t)$ (Why?). Is there anyone who would like to answer? Thanks.