I was given the following as an execrise:
Prove that the tensor product of complex line bundles over $X$ satisfies the following relationship: $$c_{1}(\otimes E_{i})=\sum c_{1}(E_{i})$$
It is not clear to me how to solve this via the defining axioms of Chern classes. My personal guess is this should hold for other characteristic classes except the Pontrajin class (whose value on line bundles should be $0$). Since the problem should be simple, I decide to ask for a hint at here.
My thought on other approaches are as follows: given $E_{i}\rightarrow X$ we can form a classifying bundle $E'_{i}\rightarrow BU_{1}$ with a classifying map $X\rightarrow BU_{1}$ such that associated principal bundle of $E_{i}$ is the pull back of $E_{i}'$. Thus it is suffice to prove this for bundles over $BU_{1}$, but any line bundles should be the pull back of univeral line bundle over $BU_{1}$. So it suffice to prove this for universal line bundles over $BU_{1}$ and its pull backs to itself in general. However even this is not clear to me.
The way I recall we proved the Whitney sum formula is by making use the diagonal map $$B\iota:BU_{i}\times BU_{j}\rightarrow BU_{i+j}$$ where $U_{i},U_{j}$ are now placed on the diagonal blocks to $U_{i+j}$ and this map is the one inverted by taking the classifying space. However it is not clear to me how the map $$\chi:U_{1}\times U_{1}\rightarrow U_{1}: (z,w)\rightarrow zw$$ representing the tensor product works on the level of classifying spaces. Therefore I do not know how to describe the map $$B\chi: BU_{1}\times BU_{1}\rightarrow BU_{1}$$ at the level of cohomology rings. It seems to me that $B\chi^{*}$ is not injective in general.
I was given the hint that I should consider the isomorphism $$B(U_{1}\times U_{1})\cong BU_{1}\times BU_{1}$$ which I used implicitly in the proof above. The professor asked me to prove this myself and suggested using Kunneth formula afterwards. But still this is not immediate to me.
It would seem that Hatcher's book on vector bundles and K-theory provides an answer at page 72. The proposition reads as follows:
The function $ w_1: Vect^1(X) \longrightarrow H^1(X;Z_2) $ is an isomorphism if $ X $ has the homotopy type of a CW complex. The same is also true for $ c_1 $ [...]
As you suggested the proof treats the universal bundle case, and then uses pullback for the general case.