Tangent Space of Product Manifold

I don't understand what you are trying to do and here's what I would do. Consider the canonical projections $\pi_X,\pi_Y$ from $X \times Y$ to $X$ and $Y$ respectively. Let $(p,q) \in X \times Y$ and now consider the map

$$f : T_{(p,q)}(X \times Y) \to T_p X \times T_q Y$$

that sends a vector $v$ to elements $\Big(d(\pi_X)_{(p,q)}(v), d(\pi_Y)_{(p,q)}(v) \Big)$. You can easily check that this map is a linear map that is an isomorphism with the inverse given by $g : T_p X \times T_q Y \to T_{(p,q)} (X \times Y)$ that sends a pair of vectors $(v,w)$ to $d(\iota_X)_p(v) +d(\iota_Y)_q(w)$, where $\iota_X : X \to X\times Y$ sends $X$ to the slice $X \times \{q\}$ and similarly for $\iota_Y$.

I'm assuming you want to show that there is an isomorphism between the two spaces, if so, I would refer you Loring Tu's excellent book introduction to manifolds. Problem 8.7 is the question you are looking for and the solution is at the back. In a nutshell, he showed an isomorphism by sending basis elements to basis elements using the differential of the projection maps.

(Tu's book can be found using Google, just google "tu introduction to manifolds")

Hope this helps.