I was wondering if the sum $TS^{2n}\oplus TS^{2n}$ is a trivial bundle?
The same is true for spheres of odd dimension (one can find a nowhere zero section of the second bundle, add it to the first, the first becomes trivial and the rest of second bundle plus trivial bundle of rk 2 is trivial too).


It seems that one should take $2n$ sections $v=(v\_1,\dots,v\_{2n})$ of $TS^{2n}$ (for example projections of coordinate vector fields from $\mathbb R^{2n+1}$ to $S^{2n}$), $u=(u\_1,\dots,u\_{2n})$ for the second part and than perturb a little u=u+av, v=v+bu.


Nevertheless I can not prove that it works. From the other point of view I see no reasons for this bundle to be non-trivial. 



Sum of two tangent bundles of $S^{2n}$
Yes. Let $V$ be a real vector bundle whose base is a $d$-dimensional manifold or cell complex, and whose fibers are $r$-dimensional. Then (1) if $r>d$ then $V=W\oplus \epsilon$ where $\epsilon$ is a trivial rank one bundle, and (2) if $r>d+1$ then the rank $r-1$ bundle $W$ is determined up to isomorphism by $V$. In particular stably trivial bundles of rank greater than $d$ are trivial.


