Power-reducing identities · proof ·what it is and when to use it

Proof: sin³x = (3 sin x − sin 3x)/4

Reduction formula for sin³ — proved by expand sin 3x, then solve. Every line below says which rule it uses, so nothing has to be taken on faith.

The proof

The triple-angle formula is a sine sum with 3x = x + 2x.

1. sin 3x = sin(x + 2x)
   Rewrite the angle.
2. = sin x cos 2x + cos x sin 2x
   Sine of a sum. · Sine of a sum
3. = sin x(1 − 2sin²x) + cos x(2 sin x cos x)
   Both double-angle forms. · Double-angle identity for cosine (form 3), Double-angle identity for sine
4. = 3 sin x − 4 sin³x
   Collect, using cos²x = 1 − sin²x. · Pythagorean identity
5. sin³x = (3 sin x − sin 3x)/4
   Solve for sin³x.

Where the proof stops applying

Both sides are undefined at the same angles — where a denominator in the proof reaches zero (for instance cos x = 0 in a tangent or secant form). Elsewhere the argument above holds for every real angle.

The same claim, checked numerically

A proof is not the same thing as a check, and this page shows both: the reasoning above, and the first few angles fed to the same engine behind the verifier. If they ever disagreed, the data would be broken — the build would fail before publishing (accuracy policy).

Related

• How would I find sin³x = (3 sin x − sin 3x)/4 myself? — the derivation, which is a different question from the proof.
• Reduction formula for sin³: when to use it — the practical side.
• All power-reducing identities · proof index

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