Power-reducing identities · trigonometric identity
sin²x = (1 − cos 2x)/2 — Power-reducing formula for sin²
When to use it
Almost always integration: ∫sin²x dx is not reachable as a power, but (1 − cos 2x)/2 integrates in two terms. Also for averaging sin² over a period.
Why it is true
The double-angle identity cos 2x = 1 − 2sin²x contains sin²x alone; solve for it and the power disappears, replaced by a cosine of twice the angle.
The full line-by-line version is on the proof page for power-reducing formula for sin²; the “how would I find this myself” version is in the derivation.
Where it comes from
Powers are eliminated by doubling the angle. General rule of thumb in this family: a squared trig function becomes a constant plus a cosine of DOUBLE the angle. Cube powers become a difference of sines of x and 3x.
- Need to lose sin²?Set-up.
- Find the double-angle formula containing sin²
- Solve for sin²Algebra.
Worked examples
- ∫ sin²x dxx/2 − sin 2x/4 + 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.
| Angle | Left side | Right side | Agree |
|---|---|---|---|
| 0° | 0 | 0 | yes |
| 7.5° | 0.017037 | 0.017037 | yes |
| 15° | 0.066987 | 0.066987 | yes |
| 18° | 0.095492 | 0.095492 | yes |
| 22.5° | 0.146447 | 0.146447 | yes |
| 30° | 0.25 | 0.25 | yes |
| 37° | 0.362181 | 0.362181 | yes |
| 45° | 0.5 | 0.5 | yes |
| 53° | 0.637819 | 0.637819 | yes |
| 60° | 0.75 | 0.75 | yes |
| 67.5° | 0.853553 | 0.853553 | yes |
| 75° | 0.933013 | 0.933013 | yes |
| 90° | 1 | 1 | yes |
| 120° | 0.75 | 0.75 | yes |
| 135° | 0.5 | 0.5 | yes |
| 150° | 0.25 | 0.25 | yes |
| 180° | 0 | 0 | yes |
| 210° | 0.25 | 0.25 | yes |
| 240° | 0.75 | 0.75 | yes |
| 270° | 1 | 1 | yes |
| 300° | 0.75 | 0.75 | yes |
| 330° | 0.25 | 0.25 | yes |
| 360° | 0 | 0 | yes |
The mistake students make
Losing the factor 1/2 on the cosine term, or writing cos²(2x). The argument doubles, the amplitude halves.
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
- cos²x = (1 + cos 2x)/2Power-reducing formula for cos²detailsproof
- cos 2x = 1 − 2 sin²xDouble-angle identity for cosine (form 3)detailsproof
- sin²θ + cos²θ = 1Pythagorean identitydetailsproof
- sin³x = (3 sin x − sin 3x)/4Reduction formula for sin³detailsproof
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