Sum-to-product identities · trigonometric identity
sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2) — Sum to product: sin A − sin B
When to use it
Differences of sines — and the limit of (sin A − sin B)/(A − B) as B → A, which is how the derivative of sine is obtained without the difference quotient of a single function.
Why it is true
Same substitution, but subtract the expansions: now the sin u cos v terms cancel and 2 cos u sin v survive.
The full line-by-line version is on the proof page for sum to product: sin a − sin b; the “how would I find this myself” version is in the derivation.
Where it comes from
Keep the half-difference where the change is. The term with (A − B)/2 is the 'small' one; that is the factor that vanishes when A = B, which is a good sanity check.
- Check A = B: the right side must be 0sin 0 = 0 kills the product ✓.
- So sin((A − B)/2) has to be a factorStructural necessity.
- The other factor is the cosine of the averageSymmetry of the two angles.
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° | -0.156434 | -0.156434 | yes |
| 7.5° | -0.090171 | -0.090171 | yes |
| 15° | -0.025196 | -0.025196 | yes |
| 18° | 0 | 0 | yes |
| 22.5° | 0.036566 | 0.036566 | yes |
| 30° | 0.093263 | 0.093263 | yes |
| 37° | 0.140066 | 0.140066 | yes |
| 45° | 0.184608 | 0.184608 | yes |
| 53° | 0.217933 | 0.217933 | yes |
| 60° | 0.236705 | 0.236705 | yes |
| 67.5° | 0.245079 | 0.245079 | yes |
| 75° | 0.240551 | 0.240551 | yes |
| 90° | 0.190983 | 0.190983 | yes |
| 120° | -0.067555 | -0.067555 | yes |
| 135° | -0.265263 | -0.265263 | yes |
| 150° | -0.494522 | -0.494522 | yes |
| 180° | -0.987688 | -0.987688 | yes |
| 210° | -1.413545 | -1.413545 | yes |
| 240° | -1.643171 | -1.643171 | yes |
| 270° | -1.587785 | -1.587785 | yes |
| 300° | -1.224393 | -1.224393 | yes |
| 330° | -0.604528 | -0.604528 | yes |
| 360° | 0.156434 | 0.156434 | yes |
The mistake students make
Swapping cosine and sine. For a difference of sines the cosine carries the average and the sine carries the half-difference.
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
- sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2)Sum to product: sin A + sin Bdetailsproof
- cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2)Sum to product: cos A − cos Bdetailsproof
Category hub: Sum-to-product identities · all identities: /identities