Need to know how to compute the transition map of $h = g^{-1}\circ f$
The unit circle has charts $f(s) = (\cos(s),\sin(s))$ is element of $\Bbb R^2$, for $-\pi < s < \pi$, and
$g(t) = \left(\cfrac{2t}{t^2 + 1}, \cfrac{t^2 - 1}{t^2 + 1}\right)$ for $t\in\Bbb R$. Compute the transition map of $h = g^{-1}\circ f$
Sorry for my poor structure of the question. Thanks in advance.