Half-angle identities · proof ·what it is and when to use it
Proof: cos(x/2) = ±√((1 + cos x)/2)
Half-angle identity for cosine — proved by solve cos 2u for cos u. Every line below says which rule it uses, so nothing has to be taken on faith.
The proof
Same rearrangement with the other double-angle form.
- cos 2u = 2cos²u − 1Double-angle form 2. · Double-angle identity for cosine (form 2)
- cos²u = (1 + cos 2u)/2Solve for cos²u.
- cos u = ±√((1 + cos 2u)/2)Root with a sign to be decided.
- u = x/2 → cos(x/2) = ±√((1 + cos x)/2)Rename.
This is the identity our own test caught: the + version is not unconditional. On (−180°, 180°) it holds; at x = 270° it does not.
Where the proof stops applying
Requires x between -180° and 180° — the + root holds for −180° < x < 180° (x/2 in the right half-plane). At x = 270°, for instance, cos 135° = −√2/2 while the + root gives +√2/2 — the identity fails, which our sampler verifies.
This is not a technicality: outside the interval, the statement as written is false, and the sign has to be chosen by hand.
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° | 0.997859 | 0.997859 | yes |
| 15° | 0.991445 | 0.991445 | yes |
| 18° | 0.987688 | 0.987688 | yes |
| 22.5° | 0.980785 | 0.980785 | yes |
| 30° | 0.965926 | 0.965926 | yes |
Related
- How would I find cos(x/2) = ±√((1 + cos x)/2) myself? — the derivation, which is a different question from the proof.
- Half-angle identity for cosine: when to use it — the practical side.
- All half-angle identities · proof index