  This note is about the observation that the various transition formulas
between bases of trigonometric polynomials can be expressed in terms binomial
coefficients. More specifically, we write the entries of the Chebyshev matrices
$ T$ and $ U$ in terms of binomial coefficients. We remark that the inverses of
certain submatrices of the Chebyshev matrix $ T$ correspond to classical power
reduction formulas for trigonometric functions. Therefore the entries of $
T^{-1}$ are expressible in terms of binomial coefficients and powers of two.
The inverses of certain submatrices of the Chebyshev matrix $ U$ appear in
power reduction formulas involving the Catalan triangles. We could not spot
them in the literature. As a corollary the Catalan triangles are interpreted as
inverses of rather natural matrices of binomial coefficients. We explain the
matrix inversions in terms of Riordan arrays. We solve the integral $ \int
_{0}^{2\pi }\cos^{2m}( x)\sin^{2n}( x) dx$ and deduce a hypergeometric identity
from the Fourier expansion of $\cos^{2m}(x)$ and $ \sin^{2n}( x)$. As a
corollary we prove that super Catalan numbers are integers. We emphasize that
from the point of view of these base changes between the trigonometric
polynomials it is more natural to work with $2\cos(x)$ and $2\sin(x)$ instead
of $\cos(x)$ and $\sin(x)$, since the transition matrices become invertible
over the integers. We do a similar analysis for the spread polynomials. This
enables us to make a conjecture of Goh and Wildberger more precise. In this
exposition we give a mostly self-contained account of the matter, the
prerequisites just being four semesters of calculus, linear algebra, elementary
complex variables and the residue theorem.
