Power-reducing identities · derivation ·proof

Where sin³x = (3 sin x − sin 3x)/4 comes from

Odd powers keep the same shape. Every odd power of sine reduces to a sum of sines of odd multiples of x. The pattern is worth knowing even if you re-derive the coefficients.

Building it step by step

1. sin¹: sin x
   Base case.
2. sin³: (3 sin x − sin 3x)/4
   This identity.
3. sin⁵: (10 sin x − 5 sin 3x + sin 5x)/16
   Next step of the same pattern — re-derive rather than memorise.

How to recall it under pressure

Re-derive it from the first line rather than searching memory: start from sin(x)**3 and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the reduction formula for sin³ page.

sampled at 800 random anglesSource: OpenStax Precalculus, Ch. 7.3 · Paul's Online Math Notes · revised 2026-09-26 ·how we check ·accuracy policy ·report an error