Degree proof of the hairy ball theorem

A nowhere-zero tangent vector field on $S^n$ can be normalized to a unit field $w(x)$ perpendicular to $x$. Then $\cos (\pi t)x+\sin (\pi t)w(x)$ is a homotopy from the identity to the antipodal map. Their mapping degrees are $1$ and $(-1{)}^{n+1}$, respectively, contradicting homotopy invariance of mapping degree when $n$ is positive and even. For $S^0$ the tangent spaces are zero, so every tangent field is zero directly.