Normally I would try to present this as a solution to a concrete problem, but I don't currently know what an intuitive reason to derive Chebyshev polynomials would be.
Normally Chebyshev polynomials
are defined as T_n(cos t) =
cos(nt), and the properties of cos are used to derive the
actual polynomials T_n(x). I think that these properties of
cos are actually best understood as properties of rotation
matrices, so what happens if we try to derive Chebyshev polynomials directly
using rotation matrices?
Let C be the class of rational polynomials in three variables,
x, y, and r, with two polynomials being
considered equal if their difference is a multiple of
x^2 + y^2 - r^2, so that in particular,
x^2 + y^2 = r^2, allowing us to think of r as the
radius of the vector [x, y].
Now let M be the class of 2x2 matrices whose entries are all
elements of C, and consider the matrix R = [x, -y; y, x]
which appears to represent an arbitrary rotation-dilation matrix
xI + yJ, and whose determinant is r^2. Raising
R to powers, we get:
R = xI + yJ
R^2 = (2x^2 - r^2)I + 2xyJ
R^3 = (4x^3 - 3xr^2)I + (4x^2 - r^2)yJ
R^4 = (8x^4 - 8x^2r^2 + r^4)I + (8x^3 - 4xr^2)yJ
Evaluating with r = 1 we can see R^n = T_n(x)I + U_n(x)yJ.