Double-angle identities · trigonometric identity
cos 2x = 1 − 2 sin²x — Double-angle identity for cosine (form 3)
When to use it
Sine-only version. It is the one to reach for when sin² is in the way, and the parent of the sin² power-reducing formula used constantly in calculus.
Why it is true
Replace sin²x in cos²x − sin²x by 1 − sin²x: the cosine disappears and 1 − 2sin²x is left.
The full line-by-line version is on the proof page for double-angle identity for cosine (form 3); the “how would I find this myself” version is in the derivation.
Where it comes from
Solve for sin². Rearranging gives the single most-used formula in integral trigonometry.
- cos 2x = 1 − 2sin²xStart.
- 2sin²x = 1 − cos 2xMove the sine term to the left.
- sin²x = (1 − cos 2x)/2Power-reducing formula. · Power-reducing formula for sin²
Worked examples
- Derive the double-angle formulas from the sum formulasSet B = A in each sum formula; cosine then yields two further forms via sin² + cos² = 1.
- Why does cos 2x have three formulas and how do you choose?They are one formula plus the Pythagorean identity; choose by which square is already in your expression.
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° | 0.965926 | 0.965926 | yes |
| 15° | 0.866025 | 0.866025 | yes |
| 18° | 0.809017 | 0.809017 | yes |
| 22.5° | 0.707107 | 0.707107 | yes |
| 30° | 0.5 | 0.5 | yes |
| 37° | 0.275637 | 0.275637 | yes |
| 45° | 0 | 0 | yes |
| 53° | -0.275637 | -0.275637 | yes |
| 60° | -0.5 | -0.5 | yes |
| 67.5° | -0.707107 | -0.707107 | yes |
| 75° | -0.866025 | -0.866025 | yes |
| 90° | -1 | -1 | yes |
| 120° | -0.5 | -0.5 | yes |
| 135° | -0 | -0 | yes |
| 150° | 0.5 | 0.5 | yes |
| 180° | 1 | 1 | yes |
| 210° | 0.5 | 0.5 | yes |
| 240° | -0.5 | -0.5 | yes |
| 270° | -1 | -1 | yes |
| 300° | -0.5 | -0.5 | yes |
| 330° | 0.5 | 0.5 | yes |
| 360° | 1 | 1 | yes |
The mistake students make
Sign slip: it is 1 − 2sin²x, not 2sin²x − 1 (that is −cos 2x). Check with x = 0: cos 0 = 1, and 1 − 0 = 1 ✓.
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 double-angle identities.
Related identities
- cos 2x = cos²x − sin²xDouble-angle identity for cosine (form 1)detailsproof
- cos 2x = 2 cos²x − 1Double-angle identity for cosine (form 2)detailsproof
- sin²x = (1 − cos 2x)/2Power-reducing formula for sin²detailsproof
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