I was thinking of the atlas of the circle $S^{1}$, I could not understand the solution in the book by Khan.
In the images attached - 
Why was $f_{U}$ taken different from $f_{V}$ ? it could have been taken the same function? is it not?
Also it is hard for me to think how it defined $f_{U} o f_{V}^{-1}$ in this way? I got the identity part but how to get $id - 2 \pi$ on $(\pi, 2 \pi ) $ ?
$(-1,0)\not\in$ domain of $f_U$, i.e. the angle $-\pi$ is excluded. If you take the same definition for $f_V$, as this time $(-1,0)\in$ domain of $f_V$, you can't define $f_V(-1,0)$ because $-\pi\not\in(-\pi,\pi)$.