In a proof I need the following fact:
Let $M$ be a manifold and $TM$ its tangent bundle. Then $M$ and $TM$ are homotopy equivalent and therefore have the same homotopy groups (and explicitly for the sphere $S^n$ we have that $\forall 0 \leq k < n:$ $\pi_k(TS^n) = \pi_k(S^n) = 0$).
I don't necessarily need an exact proof of this claim, more of a sketch. Thank you a lot in advance!
I found the answer to the question I duplicated not enlightening for my purposes so I would like to keep the question...