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.

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

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?

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

Thank you