Power-reducing identities · proof ·what it is and when to use it
Proof: cos³x = (3 cos x + cos 3x)/4
Reduction formula for cos³ — proved by expand cos 3x, then solve. Every line below says which rule it uses, so nothing has to be taken on faith.
cos³x = (3 cos x + cos 3x)/4
The proof
Same route as the sine case with the cosine sum formula.
- cos 3x = cos(x + 2x)Rewrite.
- = cos x cos 2x − sin x sin 2xCosine of a sum. · Cosine of a sum
- = cos x(2cos²x − 1) − sin x(2 sin x cos x)Double-angle forms 2 and 1. · Double-angle identity for cosine (form 2), Double-angle identity for cosine (form 1)
- = 4cos³x − 3cos xCollect, replacing sin²x with 1 − cos²x. · Pythagorean identity
- cos³x = (3 cos x + cos 3x)/4Solve.
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).
| Angle | Left side | Right side | Agree |
|---|---|---|---|
| 0° | 1 | 1 | yes |
| 7.5° | 0.974554 | 0.974554 | yes |
| 15° | 0.901221 | 0.901221 | yes |
| 18° | 0.860239 | 0.860239 | yes |
| 22.5° | 0.788581 | 0.788581 | yes |
| 30° | 0.649519 | 0.649519 | yes |
Related
- How would I find cos³x = (3 cos x + cos 3x)/4 myself? — the derivation, which is a different question from the proof.
- Reduction formula for cos³: when to use it — the practical side.
- All power-reducing identities · proof index