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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:

enter image description here

The question also has a part of its answer(without the proof of the existence of $g$) here:

Prove that $\deg_2 (f) \equiv q \mod 2$

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  • $\begingroup$ Do you know some covering maps theory, e.g. path lifting theorem? $\endgroup$ – positrón0802 Dec 10 '18 at 19:42

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