Double-angle identities · trigonometric identity
cos 2x = cos²x − sin²x — Double-angle identity for cosine (form 1)
When to use it
The symmetric form — use it when both sin² and cos² are already present and you want to collapse them into one cosine.
Why it is true
Cosine sum formula with A = B = x: cos 2x = cos x cos x − sin x sin x.
The full line-by-line version is on the proof page for double-angle identity for cosine (form 1); the “how would I find this myself” version is in the derivation.
Where it comes from
From the symmetric form to the other two. The other two cosine forms are this one plus the Pythagorean identity, used to eliminate whichever square you do not want.
- cos²x = 1 − sin²xPythagorean identity. · Pythagorean identity
- Substitute into cos²x − sin²xEliminate cosine.
- cos 2x = 1 − 2 sin²xThird form. · Double-angle identity for cosine (form 3)
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
Believing there is one double-angle cosine formula. There are three equivalent shapes and choosing the wrong one costs extra work — see the other two records.
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 = 2 cos²x − 1Double-angle identity for cosine (form 2)detailsproof
- cos 2x = 1 − 2 sin²xDouble-angle identity for cosine (form 3)detailsproof
- cos(A + B) = cos A cos B − sin A sin BCosine of a sumdetailsproof
- sin²x = (1 − cos 2x)/2Power-reducing formula for sin²detailsproof
Category hub: Double-angle identities · all identities: /identities