Transcendental functions and mechanical theorem proving in elementary geometries

We present a new approach to dealing with certain problems about trigonometric and hyperbolic function using Wu's method by transforming the two kinds of transcendental functions into polynomials. This approach has been used for mechanical theorem proving in Euclidean and several Non-Euclidean geometries. Based on our approach, several meta-theorems are established for four Non-Euclidean geometries. These meta-theorems assert that certain kinds of geometry statements that can be confirmed by our approach in one geometry can also be confirmed in three other geometries. We also proved a meta-theorem which permits us to obtain all hyperbolic identities from trigonometric identities through certain kinds of transformation and vice versa.The revision was supported by the NSF grant CCR-8702108.