Power-reducing identities · trigonometric identity

cos³x = (3 cos x + cos 3x)/4 — Reduction formula for cos³

When to use it

Cubic cosine — the sign is '+' here where the sine version is '−'. Used in integrals of cos³x and when analysing the third harmonic.

Why it is true

From cos 3x = 4cos³x − 3cos x (triple-angle formula) and solving for cos³x.

The full line-by-line version is on the proof page for reduction formula for cos³; the “how would I find this myself” version is in the derivation.

Where it comes from

Derive it at x = 0 sanity check. If you cannot remember the sign, test the formula at x = 0 where all three cosines are known.

1. Left at x = 0: 1
   cos 0 = 1.
2. (3 + 1)/4 = 1
   Plus sign works.
3. (3 − 1)/4 = 1/2
   Minus sign does not.

Worked examples

• ∫ cos³x dx
  sin x − sin³x/3 + C

Checked at these angles

Two sides of the identity evaluated with the same engine that runs the verifier. A single line disagreeing would break the build.

The mistake students make

Try it

• Verify this identity — both sides are pre-filled; change one character and watch the counterexample appear.
• Ask “which identity should I use?” about the left-hand side — this identity is what the chooser should rank first.
• Cheat sheet entry for power-reducing identities.

Related identities

• sin³x = (3 sin x − sin 3x)/4Reduction formula for sin³detailsproof
• cos²x = (1 + cos 2x)/2Power-reducing formula for cos²detailsproof
• cos 2x = cos²x − sin²xDouble-angle identity for cosine (form 1)detailsproof

Category hub: Power-reducing identities · all identities: /identities

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