In order to prove the existence of the function $g$ in the question I want to proof that the following function is a diffeomorphism (I was told a hint that it is a diffeomorphism):
$$p(t) = (\cos t, \sin t)$$
But actually I see that this function is not 1-1, and what about its inverse how can I calculate it? could anyone help me in this?
The question I am speaking about is given below:
The question also has a part of its answer(without the proof of the existence of $g$) here: