Odd Angles Formula Trivia

[Albert Chan]

We can show above sign flipping behavior, using Chebyshev Polynomial

By definition, T_n(cos(x)) = cos(n*x)

T_n(cos(pi/2-x)) = cos(n*(pi/2-x)) = cos((n-1)*pi/2 + (pi/2-n*x))
T_n(sin(x)) = sin(n*x + (n-1)*pi/2)

if n=4k+1, (n-1)*pi/2 ≡ 0 (mod 2*pi), then T_n(sin(x)) = sin(n*x+0) = sin(n*x)
if n=4k−1, (n-1)*pi/2 ≡ pi (mod 2*pi), then T_n(sin(x)) = sin(n*x+pi) = - sin(n*x)

QED

Extend above for even n's:

if n=4k+0, n*pi/2 ≡ 0 (mod 2*pi), then T_n(sin(x)) = cos(0-n*x) = cos(n*x)
if n=4k+2, n*pi/2 ≡ pi (mod 2*pi), then T_n(sin(x)) = cos(pi-n*x) = - cos(n*x)