Homotopic functions in $n$-sphere

Question

Can anyone give me a hint in this problem? Please. Let $X$ be a topological space. If $f,g$ are continuous function of $f,g \colon X\to S^n$ which $f(x)\neq -g(x)$ for all $x \in X $, show $f$ and $g$ are homotopics.

Answer 1

$f(x)+g(x)\neq 0$ implies that $c(t)=tf(x)+(1-t)g(x)\neq 0, t\in [0,1]$. In fact, $c(t)=0$ implies that $tf(x)+(1-t)g(x)=0$ and $-tf(x)=(1-t)g(x)$ taking the norm of the both sides of the last equation, we deduce that $t=1-t, t=1/2$ and $f(x)/2+g(x)/2=0$. Contradiction.

$H(t,x)={{c(t)}\over{\|c(t)\|}}$ is the requested homotopy.