Tangent Bundle of Product is diffeomorphic to Product of Tangent Bundles

[Amateur659]

My apologies if this question is trivial. I have searched the forum and haven't found an existing answer to this question.

I've been working through differential geometry problem sets I found online (associated with MATH 481 at UIUC) and am struggling to show that T(MxN) is diffeomorphic to TM x TN.

My intuition is that I could set up the product of projection maps from MxN to M and MxN to N to form the diffeomorphism. However, I seem to be stuck on the implementation (which probably indicates gaps in my understanding of previous material).

Thanks for your assistance.

 

[fresh_42]

This is the kind of question that requires (only) a precise application of all definitions. The sets are clearly identical:
$$
T(M\times N)= \bigcup_{(p,q)\in M\times N}\{(p,q)\} \times T_{(p,q)}(M\times N) = \bigcup_{p\in M} \{p\}\times T_{p}(M)\times \bigcup_{q\in N} \{q\}\times T_{q}(N)=T(M) \times T(N)
$$
Next, we need a differentiable structure on both. Differentiability is a local property. So we can pick a point $(p,q)\in U\times V$ from a contractible neighborhood. By defining it as $U\times V$ we already fixed the topology, namely the product topology. Now gather all local diffeomorphisms $TU=U\times \mathbb{R}^n\, , \,TV=V\times \mathbb{R}^m$ and combine them to a diffeomorphism $T(U\times V)= U\times V \times \mathbb{R}^{n+m}.$