Chebyshev polynomial crop up in virtually every area of Numerical analysis and they hold
particular importance in recent advancement in subject such as Orthogonal. In this paper, we
present the classical theory of Chebyshev polynomial starting from the definition and generation
of the family of both the 1st, the 2nd kind and the complex classical polynomial of Chebyshev
polynomial, which are related to their real and imaginary parts. This permit one to develop the
theory of a class of Orthogonal polynomial that are that basis for fitting non-algebraic function
with polynomial of maximum efficiency. The generated series polynomial is use to Economize
the approximation of some One-step methods such as RKMII and Runge-kutta-Feldberg method.
An Economic table that represent the operation at various states of mesh point is shown.