Odd Angles Formula Trivia

[Albert Chan]

(12-20-2018 05:38 PM)Albert Chan Wrote:  Challenge: prove above pattern continues: if sin(4nx ± x) = f(s,c), then cos(4nx ± x) = ± f(c,s)

Let angle A = 4kx + x, s = sin(x), c = cos(x):

sin(A+2x) = sin(A) cos(2x) + cos(A) sin(2x) = f(s,c) (c² - s²) + f(c,s) (2 s c) = g(s,c)
cos(A+2x) = cos(A) cos(2x) - sin(A) sin(2x) = f(c,s) (c² - s²) - f(s,c) (2 s c) = -g(c,s)

sin(A+4x) = sin(A+2x) cos(2x) + cos(A+2x) sin(2x) = g(s,c) (c² - s²) - g(c,s) (2 s c) = h(s,c)
cos(A+4x) = cos(A+2x) cos(2x) - sin(A+2x) sin(2x) = -g(c,s) (c² - s²) - g(s,c) (2 s c) = h(c,s)

Angles A+2x and A+4x is same as 4(k+1)x ± x
Post #1 examples already shows n=1 work, and by induction, if n=k work, n=k+1 also work.

QED