Sum-to-product identities · trigonometric identity
cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2) — Sum to product: cos A − cos B
When to use it
Differences of cosines — and the only sum-to-product formula with a leading minus sign. Forgetting that minus is the single most common error in this group.
Why it is true
Set u = (A + B)/2, v = (A − B)/2 and subtract: cos(u + v) − cos(u − v) = (cos u cos v − sin u sin v) − (cos u cos v + sin u sin v) = −2 sin u sin v. The cosine products cancel, the sine products add with a minus.
The full line-by-line version is on the proof page for sum to product: cos a − cos b; the “how would I find this myself” version is in the derivation.
Where it comes from
Sanity-check with A = B. When A = B the left side is 0 and the right side is 0 because sin 0 = 0 — consistent. Then one numeric test fixes the overall sign.
- A = B: both sides 0Necessary condition ✓.
- A = 0, B = π: left = 1 − (−1) = 2Concrete test.
- right = −2 sin(π/2) sin(−π/2) = +2Matches only with the leading minus.
Worked examples
No worked example is written for this identity yet —the examples index lists what is covered, and the pattern below shows the identity working at real angles.
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° | 0.012312 | 0.012312 | yes |
| 7.5° | 0.016103 | 0.016103 | yes |
| 15° | 0.007106 | 0.007106 | yes |
| 18° | 0 | 0 | yes |
| 22.5° | -0.014312 | -0.014312 | yes |
| 30° | -0.04752 | -0.04752 | yes |
| 37° | -0.088375 | -0.088375 | yes |
| 45° | -0.145533 | -0.145533 | yes |
| 53° | -0.2123 | -0.2123 | yes |
| 60° | -0.277146 | -0.277146 | yes |
| 67.5° | -0.351639 | -0.351639 | yes |
| 75° | -0.429536 | -0.429536 | yes |
| 90° | -0.587785 | -0.587785 | yes |
| 120° | -0.858368 | -0.858368 | yes |
| 135° | -0.940552 | -0.940552 | yes |
| 150° | -0.970554 | -0.970554 | yes |
| 180° | -0.843566 | -0.843566 | yes |
| 210° | -0.459289 | -0.459289 | yes |
| 240° | 0.12932 | 0.12932 | yes |
| 270° | 0.809017 | 0.809017 | yes |
| 300° | 1.43358 | 1.43358 | yes |
| 330° | 1.860547 | 1.860547 | yes |
| 360° | 1.987688 | 1.987688 | yes |
The mistake students make
Dropping the minus. Quick check: take A = 0, B = 60°. Left side is 1 − 0.5 = 0.5 which is positive while sin(A/2)sin(B/2) is positive too — so the sign only works out once you test it; test it.
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 sum-to-product identities.
Related identities
- cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2)Sum to product: cos A + cos Bdetailsproof
- sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2)Sum to product: sin A − sin Bdetailsproof
Category hub: Sum-to-product identities · all identities: /identities