Double-angle identities · proof ·what it is and when to use it
Proof: cos 2x = 2 cos²x − 1
Double-angle identity for cosine (form 2) — proved by eliminate sine from form 1. Every line below says which rule it uses, so nothing has to be taken on faith.
cos 2x = 2 cos²x − 1
The proof
One substitution.
- cos 2x = cos²x − sin²x
- sin²x = 1 − cos²xPythagorean identity. · Pythagorean identity
- cos 2x = cos²x − (1 − cos²x)Substitute.
- cos 2x = 2cos²x − 1Simplify.
Where the proof stops applying
Both sides are undefined at the same angles — where a denominator in the proof reaches zero (for instance cos x = 0 in a tangent or secant form). Elsewhere the argument above holds for every real angle.
The same claim, checked numerically
A proof is not the same thing as a check, and this page shows both: the reasoning above, and the first few angles fed to the same engine behind the verifier. If they ever disagreed, the data would be broken — the build would fail before publishing (accuracy policy).
| 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 |
Related
- How would I find cos 2x = 2 cos²x − 1 myself? — the derivation, which is a different question from the proof.
- Double-angle identity for cosine (form 2): when to use it — the practical side.
- All double-angle identities · proof index