Odd Angles Formula Trivia

[Albert Chan]

The same trick work for sin(x)ⁿ, but only odd n can convert to multiple angle sin

Example, for sin(x)³:

(z - 1/z) = 2i sin(x)
(z - 1/z)³ = z³ - 3z + 3/z - 1/z³ = (z³ - 1/z³) - 3(z - 1/z)

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

For even n, we get back linear combinations of multiple angles cos:

(z - 1/z)⁶ = z⁶ - 6z⁴ + 15z² - 20 + 15/z² - 6/z⁴ + 1/z⁶ = (z⁶ + 1/z⁶) - 6(z⁴ + 1/z⁴) + 15(z² + 1/z²) - 20

Divide above (both side) by -2, we get: 2⁵ sin(x)⁶ = -cos(6x) + 6 cos(4x) - 15 cos(2x) + 20/2

Update: we can deduce the signs without z's. From right to left, sign pattern is + − + − + − ...