I have been trying to solve this exercise from the book Fundamental Group and Covering Spaces written by Elon Lages Lima, chapter 3. It says that given $f,g : B^{2} \to S^{2}$ continuous such that for $(x,y) \in S^{1}$, $f(x,y) = (x,y,0)$ and $g(x,y) = (-y,x,0)$, there exists $(x,y) \in B^{2}$ with $f(x,y) = g(x,y)$ or $f(x,y) = -g(x,y)$.
I'm not really sure how to proceed. The definition of $f,g$ makes me think that I should try inner product or projective space. Or maybe i should simply proceed by contradiction and try to construct a function in contradiction with some application like Borsuk-Ulam theorem or that doesn't exists a not null tangent vectorial field on S^{2}. Neither of those have worked to me.
Any ideas, suggestions or answers? :)
If neither $f(x,y)=g(x,y)$ nor $f(x,y)=-g(x,y)$ then there is a well defined nonzero tangent vector to the sphere pointing along the great circle containing $f(x,y)$ and $g(x,y)$ in the direction from $f(x,y)$ toward $g(x,y)$. This uses that $f(x,y) \neq g(x,y)$ to guarantee nonzero length, and $f(x,y) \neq -g(x,y)$ to guarantee there is only one such direction.
So if you can fill in from your assumptions about $f,g$ on the equator of the sphere to this vector field, you can finish with Borsuk-Ulam as you suggest. Thus the idea you had might be made to work.
EDIT: I think the way is to define $F(x,y,z)=f(x,y)$ and $G(x,y,z)=g(x,y),$ so that now $F,G$ are maps from the sphere $S^2$ to itself, where the sphere is considered as points in three space where $x^2+y^2+z^2=1$. In other words to get the image of a point on the sphere under $F$ or $G$ one projects onto the $xy$ plane and reads off the value of $f$ or $g$ respectively. This done we now have the functions $F,G$ defined on $S_2$ and I think the above idea produces a nonvanishing vector field.