Sum-to-product identities · trigonometric identity
cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2) — Sum to product: cos A + cos B
When to use it
Sums of two cosines, commonly when combining two waves of similar frequency: the product form makes the amplitude modulation visible.
Why it is true
Add the two cosine expansions with the u, v substitution: the sin u sin v terms cancel and 2 cos u cos v remain.
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
Both cosines, both stay cosines. Mnemonic: with a sum of cosines, nothing changes function; with a difference of cosines, everything does.
- cos + cos → cos·cosPattern.
- cos − cos → sin·sin with a minusThe companion formula. · Sum to product: cos A − cos B
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° | 1.987688 | 1.987688 | yes |
| 7.5° | 1.966787 | 1.966787 | yes |
| 15° | 1.924746 | 1.924746 | yes |
| 18° | 1.902113 | 1.902113 | yes |
| 22.5° | 1.862071 | 1.862071 | yes |
| 30° | 1.779571 | 1.779571 | yes |
| 37° | 1.685646 | 1.685646 | yes |
| 45° | 1.559747 | 1.559747 | yes |
| 53° | 1.415931 | 1.415931 | yes |
| 60° | 1.277146 | 1.277146 | yes |
| 67.5° | 1.117006 | 1.117006 | yes |
| 75° | 0.947174 | 0.947174 | yes |
| 90° | 0.587785 | 0.587785 | yes |
| 120° | -0.141632 | -0.141632 | yes |
| 135° | -0.473661 | -0.473661 | yes |
| 150° | -0.761497 | -0.761497 | yes |
| 180° | -1.156434 | -1.156434 | yes |
| 210° | -1.272762 | -1.272762 | yes |
| 240° | -1.12932 | -1.12932 | yes |
| 270° | -0.809017 | -0.809017 | yes |
| 300° | -0.43358 | -0.43358 | yes |
| 330° | -0.128496 | -0.128496 | yes |
| 360° | 0.012312 | 0.012312 | yes |
The mistake students make
Reaching for a sine factor. Cosine plus cosine gives cosine times cosine — both functions stay cosines.
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 sin((A + B)/2) sin((A − B)/2)Sum to product: cos A − cos Bdetailsproof
- sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2)Sum to product: sin A + sin Bdetailsproof
- cos A cos B = ½[cos(A − B) + cos(A + B)]Product to sum: cos A cos Bdetailsproof
Category hub: Sum-to-product identities · all identities: /identities