Product-to-sum identities · trigonometric identity
sin A sin B = ½[cos(A − B) − cos(A + B)] — Product to sum: sin A sin B
When to use it
Products of two sines — and, with A = B, the power-reducing formula sin²A = ½[1 − cos 2A]. Also the orthogonality relation Fourier series are built on.
Why it is true
Subtract the two cosine expansions: cos(A − B) − cos(A + B) = (cos A cos B + sin A sin B) − (cos A cos B − sin A sin B) = 2 sin A sin B, so halving the difference of the cosines gives back the product.
The full line-by-line version is on the proof page for product to sum: sin a sin b; the “how would I find this myself” version is in the derivation.
Where it comes from
A = B gives sin². Specialising this formula is the usual way to reach the sin² reduction.
- B = ASpecialise.
- cos 0 = 1, cos 2A staysSimplify.
- sin²A = ½(1 − cos 2A)Power-reducing formula. · Power-reducing formula for sin²
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 | 0 | yes |
| 7.5° | 0.028807 | 0.028807 | yes |
| 15° | 0.073509 | 0.073509 | yes |
| 18° | 0.095492 | 0.095492 | yes |
| 22.5° | 0.132453 | 0.132453 | yes |
| 30° | 0.203368 | 0.203368 | yes |
| 37° | 0.277887 | 0.277887 | yes |
| 45° | 0.369462 | 0.369462 | yes |
| 53° | 0.46377 | 0.46377 | yes |
| 60° | 0.545007 | 0.545007 | yes |
| 67.5° | 0.62713 | 0.62713 | yes |
| 75° | 0.700658 | 0.700658 | yes |
| 90° | 0.809017 | 0.809017 | yes |
| 120° | 0.808504 | 0.808504 | yes |
| 135° | 0.687569 | 0.687569 | yes |
| 150° | 0.497261 | 0.497261 | yes |
| 180° | 0 | 0 | yes |
| 210° | -0.456773 | -0.456773 | yes |
| 240° | -0.673028 | -0.673028 | yes |
| 270° | -0.587785 | -0.587785 | yes |
| 300° | -0.310356 | -0.310356 | yes |
| 330° | -0.052264 | -0.052264 | yes |
| 360° | 0 | 0 | yes |
The mistake students make
Writing the sine pair as a sum of sines. A product of two sines becomes a difference of 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 product-to-sum identities.
Related identities
- cos A cos B = ½[cos(A − B) + cos(A + B)]Product to sum: cos A cos Bdetailsproof
- sin²x = (1 − cos 2x)/2Power-reducing formula for sin²detailsproof
- cos 2x = 1 − 2 sin²xDouble-angle identity for cosine (form 3)detailsproof
Category hub: Product-to-sum identities · all identities: /identities