Relation to trigonometric functions.

    The signal property of Chebyshev polynomials is the trigonometric representation on [-1,1].

Consider the following expansion using the Mathematica command "FunctionExpand."

Exploration 2.

We are interested in the polynomial form of  Cos[n ArcCos[x]],  however we will restrict our analysis to [-1,1].

Here is a list of several expansions.  

(c) John H. Mathews 2004