Tensor product identities in two variables are quite common in mathematics: Exponential, logarithmic, trigonometric, and hyperbolic functions all satisfy tensor product identities, and the binomial theorem is a familiar example of a tensor product identity for polynomial functions. This article presents a new elementary technique which can derive and prove all of these identities — automatically! This unified approach is based on the author's recent research on uniqueness theory for dual asymptotic expansions and remainder theory for Taylor interpolation on two lines. These results provide a simple iterative algorithm which derives a tensor product from a closed-form expression using the author's asymptotic splitting operator, and a simple hyperbolic eigenfunction criterion which proves that the two forms are identically equal. The author has implemented these methods as a complete derivation and proof system in the Maple 8 computer algebra system. The Maple code, which is surprisingly brief, is included in its entirety. The article also includes numerous examples which illustrate a variety of novel techniques for deriving and proving tensor product identities using this simple but effective system. (Maple is a registered trademark of Waterloo Maple Inc.