Double-angle identities · trigonometric identity
cos 2x = 2 cos²x − 1 — Double-angle identity for cosine (form 2)
When to use it
Cosine-only version: use it when the expression contains cos² and you want a single cosine, and as the source of the cos² power-reducing formula.
Why it is true
Replace cos²x in cos²x − sin²x with 1 − sin²x — or, equivalently, eliminate sin² with the Pythagorean identity.
The full line-by-line version is on the proof page for double-angle identity for cosine (form 2); the “how would I find this myself” version is in the derivation.
Where it comes from
Solve for cos². This is the standard route to the power-reducing formula for cosine.
- cos 2x = 2cos²x − 1Start.
- cos 2x + 1 = 2cos²xAdd 1 to both sides.
- cos²x = (1 + cos 2x)/2Divide by 2 — the power-reducing form. · Power-reducing formula for cos²
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
Writing 2cos 2x − 1 or forgetting the square on the cosine: the 2 multiplies the function value, the square is on cos x.
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 = 1 − 2 sin²xDouble-angle identity for cosine (form 3)detailsproof
- cos²x = (1 + cos 2x)/2Power-reducing formula for cos²detailsproof
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