Relation of tangential and normal characteristic numbers

Question

May states that the tangential characteristic numbers of a manifold $M$ are all $0$ if and only if the normal characteristic numbers are all $0$. His proof is that by the Whitney duality formula $$w(\tau)w(\nu)=1$$ (for $\tau$ the tangent bundle and $\nu$ the normal bundle of an embedding of $M$) all $w_i(\tau)$ are a polynomial in the $w_i(\nu)$ and converse. But we cannot easily seperate $w_i(\tau)$ in $w(\tau)w(\nu)$, can we? So how are these polynomials found?