It's known that $S^2$ is a $1$-dimensional complex manifold. Let $\varepsilon^n$ denote the trivial vector bundle of rank $n$, then $TS^2\oplus\varepsilon^1 = \varepsilon^3$, so by the Whitney product formula we have $c(\varepsilon^3)=c(TS^2\oplus\varepsilon^1) = c(TS^2)\cdot c(\varepsilon^1)$ which implies $c(TS^2) = 1$ since $c(\varepsilon^k) = 1$, where $c(E)$ denotes the total Chern classes of $E$. Hence $c_1(TS^2)=0$.
But the top Chern number is $\langle c_1(TS^2), [S^2]\rangle = \chi(S^2)\neq 0$, so what leads to this contradiction?
Wikipedia says $c_1(TS^2)\neq 0$, so maybe there is something wrong with the Whitney product formula, but I don't see it.