Solution
= Solution
The left side in part (c) is a <polynomial> in $\varepsilon$ of degree at most $n+1$. If $n$ is even, $(n+1)/2$ is a half-integer, so
$$
(1+\varepsilon^2)^{(n+1)/2}
$$
is not a <polynomial> and cannot agree with the left side on an interval. This contradiction shows that an everywhere nonzero tangent unit <vector> field $\phi$ cannot exist on an even-dimensional sphere.