Power-reducing identities · trigonometric identity
cos²x = (1 + cos 2x)/2 — Power-reducing formula for cos²
When to use it
The cosine twin: ∫cos²x dx, RMS values of a wave, and any average of cos² over a whole number of periods (the cosine term integrates to 0 and 1/2 is left).
Why it is true
Same move on the other cosine form: cos 2x = 2cos²x − 1 gives cos²x = (1 + cos 2x)/2, which is why the sign inside is opposite to the sine version.
The full line-by-line version is on the proof page for power-reducing formula for cos²; the “how would I find this myself” version is in the derivation.
Where it comes from
Check against the Pythagorean identity. sin² + cos² from these two formulas must give 1 — the cos 2x terms cancel.
- (1 − cos 2x)/2 + (1 + cos 2x)/2Add the two reduction formulas. · Power-reducing formula for sin², Power-reducing formula for cos²
- = 2/2 = 1Cosine terms cancel — consistent with sin² + cos² = 1. · Pythagorean identity
Worked examples
- Simplify sin²θ + cos²θ + 3cos²θ1 + 3cos²θ — and with the power-reducing formula, 5/2 + (3/2)cos 2θ.
- ∫₀^π cos²x dxπ/2
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.
| Angle | Left side | Right side | Agree |
|---|---|---|---|
| 0° | 1 | 1 | yes |
| 7.5° | 0.982963 | 0.982963 | yes |
| 15° | 0.933013 | 0.933013 | yes |
| 18° | 0.904508 | 0.904508 | yes |
| 22.5° | 0.853553 | 0.853553 | yes |
| 30° | 0.75 | 0.75 | yes |
| 37° | 0.637819 | 0.637819 | yes |
| 45° | 0.5 | 0.5 | yes |
| 53° | 0.362181 | 0.362181 | yes |
| 60° | 0.25 | 0.25 | yes |
| 67.5° | 0.146447 | 0.146447 | yes |
| 75° | 0.066987 | 0.066987 | yes |
| 90° | 0 | 0 | yes |
| 120° | 0.25 | 0.25 | yes |
| 135° | 0.5 | 0.5 | yes |
| 150° | 0.75 | 0.75 | yes |
| 180° | 1 | 1 | yes |
| 210° | 0.75 | 0.75 | yes |
| 240° | 0.25 | 0.25 | yes |
| 270° | 0 | 0 | yes |
| 300° | 0.25 | 0.25 | yes |
| 330° | 0.75 | 0.75 | yes |
| 360° | 1 | 1 | yes |
The mistake students make
Sign: cos² uses '+ cos 2x' while sin² uses '− cos 2x'. Adding them must give 1, which is a fast self-check.
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
- sin²x = (1 − cos 2x)/2Power-reducing formula for sin²detailsproof
- cos 2x = 2 cos²x − 1Double-angle identity for cosine (form 2)detailsproof
- cos A cos B = ½[cos(A − B) + cos(A + B)]Product to sum: cos A cos Bdetailsproof
- cos³x = (3 cos x + cos 3x)/4Reduction formula for cos³detailsproof
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