Product-to-sum identities · trigonometric identity
cos A cos B = ½[cos(A − B) + cos(A + B)] — Product to sum: cos A cos B
When to use it
Matching-function products. Set A = B and this is the power-reducing formula cos²A = ½[1 + cos 2A] — which is why it exists in a calculus course.
Why it is true
Add the two cosine expansions: the sin A sin B terms cancel and 2 cos A cos B remains.
The full line-by-line version is on the proof page for product to sum: cos a cos b; the “how would I find this myself” version is in the derivation.
Where it comes from
Specialise to get the power-reducing form. Put B = A: cos(A − A) = cos 0 = 1 and cos(A + A) = cos 2A.
- cos A cos A = ½[cos 0 + cos 2A]Set B = A. · Product to sum: cos A cos B
- cos 0 = 1Unit circle.
- cos²A = ½(1 + cos 2A)Simplify — the power-reducing formula. · Power-reducing formula for cos²
Worked examples
- Rewrite as a sum: (a) cos 2θ cos 4θ (b) sin θ sin 3θ(a) ½[cos 2θ + cos 6θ] (b) ½[cos 2θ − cos 4θ]
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° | 0.987688 | 0.987688 | yes |
| 7.5° | 0.966998 | 0.966998 | yes |
| 15° | 0.926149 | 0.926149 | yes |
| 18° | 0.904508 | 0.904508 | yes |
| 22.5° | 0.866776 | 0.866776 | yes |
| 30° | 0.791154 | 0.791154 | yes |
| 37° | 0.708398 | 0.708398 | yes |
| 45° | 0.602908 | 0.602908 | yes |
| 53° | 0.489947 | 0.489947 | yes |
| 60° | 0.388573 | 0.388573 | yes |
| 67.5° | 0.281013 | 0.281013 | yes |
| 75° | 0.178159 | 0.178159 | yes |
| 90° | 0 | 0 | yes |
| 120° | -0.179184 | -0.179184 | yes |
| 135° | -0.165071 | -0.165071 | yes |
| 150° | -0.090524 | -0.090524 | yes |
| 180° | 0.156434 | 0.156434 | yes |
| 210° | 0.352244 | 0.352244 | yes |
| 240° | 0.31466 | 0.31466 | yes |
| 270° | 0 | 0 | yes |
| 300° | -0.46679 | -0.46679 | yes |
| 330° | -0.861281 | -0.861281 | yes |
| 360° | -0.987688 | -0.987688 | yes |
The mistake students make
Expecting a minus because cos(A + B) has one. In the sum of the two expansions the minuses cancel each other.
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 product-to-sum identities.
Related identities
- sin A sin B = ½[cos(A − B) − cos(A + B)]Product to sum: sin A sin Bdetailsproof
- cos²x = (1 + cos 2x)/2Power-reducing formula for cos²detailsproof
- cos 2x = 2 cos²x − 1Double-angle identity for cosine (form 2)detailsproof
Category hub: Product-to-sum identities · all identities: /identities