Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 25J / d / Solution

The left side in part (c) is a polynomial in $\epsilon$ of degree at most $n+1$. If $n$ is even, $(n+1)/2$ is a half-integer, sois not a polynomial and cannot agree with the left side on an interval. This contradiction shows that an everywhere nonzero tangent unit vector field $\varphi$ cannot exist on an even-dimensional sphere.