This exercise is 11.13 from Bott–Tu: Use the existence of the global angular form $\psi$ to prove that if an oriented sphere bundle $E$ has a section, then its Euler class vanishes.
Now $d\psi=-\pi^*e$, so $e=-ds^*\psi$. I'd like to show that $s^*\psi$ is a cocycle. I know that $\psi$'s restriction to each $S^n$ fiber is a generator of the cohomology but I'm not sure how to use this fact. I guess this fact means that $i_p^*\psi$ generates the cohomology of $S^n$ where $i_p:S^n\to E$ is the inclusion over $p\in M$. So I was thinking of doing something like $(s\pi i_p)^*\psi$, and showing that this is closed, but I wasn't able to get this to work.