Chern classes tangent bundle $\mathbb{C}P^n$.

Question

Let $V \in Vect_k(M, \mathbb{C})$. We define Chern classes $c_i(V) \in H^{2i}(M, \mathbb{Z})$ with the usual 4 axioms. Now we consider the tangent bundle $$ \mathbb{C}^n \hookrightarrow T\mathbb{C}P^n \stackrel{p}{\longrightarrow} \mathbb{C}P^n. $$ How can I prove that the total Chern class $c(T\mathbb{C}P^n) = (x+1)^{n+1}$? We can assume that $H^*(\mathbb{C}P^n, \mathbb{Z}) \simeq \mathbb{Z}[x]/(x^{n+1})$ with $x \in H^2(\mathbb{C}P^n, \mathbb{Z})$.

Answer 1

This is proved in Milnor&Stasheff, page 168. I suggest you look it up yourself because I cannot find a better proof over the top of my head. But if you know how to prove the Stifel-Whitney classes for $\mathbb{RP}^{n}$, then you should have no difficulty to prove this case using similar axioms.