I have an exercises as follows: Let $E$ be a trivial bundle on $S^n$. Prove that the Whitney sum $TS^n\oplus E$ is also trivial. The hint is using the normal bundle of $TS^n$, but I don't know how to use it. Some one can help me? Thanks a lot!
You just notice that the Whitney sum of normal bundle and tangent bundle is trivial...As they add up to be the underlying Euclidean space of your sphere. And give a diffeomorphism between the line bundle and normal bundle. Actually you can use this to prove that the product of two spheres, one is odd dimensional, must have a trivial bundle.