Odd Angles Formula Trivia

[Albert Chan]

A trick to convert cos(x)ⁿ as linear combinations of multiple angle cos

Example: cos(x)³ to linear combinations of cos(3x) and cos(x)

Let complex number z = exp(i x)

(z + 1/z) = 2 cos(x)
(z + 1/z)³ = z³ + 3z + 3/z + 1/z³ = (z³ + 1/z³) + 3(z + 1/z)

Divide above (both side) by 2, we get: 2² cos(x)³ = cos(3x) + 3 cos(x)

We can skip above steps, and write formula directly !
Example: 6th row of pascal triangle = 1 6 15 20 15 6 1

→ 2⁵ cos(x)⁶ = cos(6x) + 6 cos(4x) + 15 cos(2x) + 20/2