Homotopic maps from $\mathbb{S}^{n} \to \mathbb{S}^{n}$

Question

I'm trying to prove that if two (continuous) maps $f, g : \mathbb{S}^{n} \to \mathbb{S}^{n}$ are such that $f(x) \neq -g(x)$ for any $x \in \mathbb{S}^{n}$, then $f$ and $g$ are homotopic. But I can't seem to have achieved any satisfactory results.

Could anyone please give me a short clue as to how to attack this problem?

I, like anyone who loves mathematics, am not asking for the complete solution. Rather, I am curious which ideas are used in solving the problem.

Thank you

Answer 1

Homotopies in $\mathbb{R}^{n+1}$ often look like $tp(x)+(1-t)q(x)$. We want to do this but restrict it to the sphere. What's the problem if $f(x)=-g(x)$?

Answer 2

Define the homotopy:

$$H(t,x) = \frac{tf(x) + (1-t)g(x)}{||tf(x) + (1-t)g(x)||}$$