How to prove that $w_1(E)=w_1(\det E)$? How to prove that the first Stiefel-Whitney class $w_1 (E)$ of a real rank $n$ vector bundle over a manifold M is equal to $w_1(\det E)$, where $\det E$ is the $n$-th wedge power of $E$?
(I want to assume the "axiomatic" definition of Stiefel-Whitney classes, as given e.g. in the book by Milnor and Stasheff).
I have just been asked an analogous question by a younger guy, but I think I could only find a proof starting from a different definition of the $w_i$'s. Perhaps I'm just missing something? Of course, feel free to close it if you find it's to homework-ish for MO standards.
 A: $E \oplus det E$ is orientable (its structure group $O(n)$ is represented in $SO(n+1)$), so its $w_1$ vanishes; and $w_1(E \oplus det E) = w_1(E) + w_1(det E)$.
A: This is really a long comment regarding some of the above discussion.
Hatcher (in his Vector Bundles notes) certainly proves that the characteristic classes defined using Leray-Hirsch satisfy the axioms from Milnor-Stasheff.  But $w_1$ is a more basic object: the axioms specify its value on the tautological bundle over ${\Bbb R}P^1$ (= $S^1$) and this immediately determines its values on all line bundles (see p. 81 of Hatcher's notes).  One can then see that as an element of 
$H^1(X; {\Bbb Z}/2) = Hom(H_1 X, {\Bbb Z}/2) = Hom(\pi_1 X, {\Bbb Z}/2)$, 
$w_1(L)$ simply answers the question: "Along which loops is L trivial?"  (Actually, this is true for all bundles, not just lines.)  From this point of view, multiplicativity ($w_1 (L\otimes K) = w_1(L) + w_1(K)$) is a quick exercise (hmmm... what should a homomorphism from a multiplicative group to an additive group be called?  Anyway, I just mean it's a homomorphism from the Picard group of line bundles to $H^1$.).  Alternatively, it follows from the H-space structure on ${\Bbb R}P^\infty$ defined via the map ${\Bbb R}P^\infty\times {\Bbb R}P^\infty \to {\Bbb R}P^\infty$ classifying $\gamma^1_\infty \otimes \gamma^1_\infty$.  This is spelled out in  my notes on vector bundles  (it's written in the complex case but works the same way in the real case).  I couldn't quickly see where Hatcher discusses this point.  
Incidentally, my notes also discuss the relationship between orientability and $w_1$.  I've never been crazy about discussions of this point in the literature (e.g. Hatcher states this relationship only for spaces of the homotopy type of a CW complex, although he doesn't seem to use that assumption in his proof).
