The question I'm about to ask has been asked several times on MSE but none with an answer that involves only degree theory.
I'd like to prove the general case of the exercise in the title without using homology theory, maybe using induction since I managed to prove the case $n=1$ through degree theory on covering spaces, proving that called $\tilde{f}$ the "covering", $\tilde{f}$ satisfies $\frac{1}{2\pi}(\tilde{f}(2\pi)-\tilde{f}(0)) = 2k+1$.
For example, for the case $n=2$, I received an hint to find a regular value whose preimage doesn't intersect the equator line and find an homotpy beetween $f$ and a map with same property.
I don't understand very well this idea so I don't even know whether this idea could be applied only to the case $n=2$ since I'm don't see where it's going.
Any help or direct proof would be appreciated