Product-to-sum identities · trigonometric identity
sin A cos B = ½[sin(A + B) + sin(A − B)] — Product to sum: sin A cos B
When to use it
A product of sine and cosine that you must integrate, sum, or compare at two frequencies. In calculus this is the standard opening move on ∫sin 3x cos x dx.
Why it is true
Add and subtract the two sine sum formulas: sin(A + B) + sin(A − B) leaves exactly 2 sin A cos B standing.
The full line-by-line version is on the proof page for product to sum: sin a cos b; the “how would I find this myself” version is in the derivation.
Where it comes from
Never memorise — re-derive in four lines. There are four product-to-sum formulas and two directions. Memorising eight lines is worse than knowing the trick: expand the sum and the difference, then add or subtract.
- Write the sum formula and the difference formula for the function you havesin here.
- Add them if the product is mixed (sin·cos)Mixed products survive addition.
- Subtract them if the product matches (sin·sin or cos·cos)Same-function products survive subtraction.
- Divide by 2Each surviving term appears twice.
Worked examples
- ∫ sin 3x cos x dx−cos 4x/8 + cos 2x/4 + C
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° | 0.127308 | 0.127308 | yes |
| 15° | 0.248161 | 0.248161 | yes |
| 18° | 0.293893 | 0.293893 | yes |
| 22.5° | 0.35903 | 0.35903 | yes |
| 30° | 0.456773 | 0.456773 | yes |
| 37° | 0.533816 | 0.533816 | yes |
| 45° | 0.602908 | 0.602908 | yes |
| 53° | 0.650182 | 0.650182 | yes |
| 60° | 0.673028 | 0.673028 | yes |
| 67.5° | 0.678426 | 0.678426 | yes |
| 75° | 0.664899 | 0.664899 | yes |
| 90° | 0.587785 | 0.587785 | yes |
| 120° | 0.310356 | 0.310356 | yes |
| 135° | 0.165071 | 0.165071 | yes |
| 150° | 0.052264 | 0.052264 | yes |
| 180° | -0 | 0 | yes |
| 210° | 0.203368 | 0.203368 | yes |
| 240° | 0.545007 | 0.545007 | yes |
| 270° | 0.809017 | 0.809017 | yes |
| 300° | 0.808504 | 0.808504 | yes |
| 330° | 0.497261 | 0.497261 | yes |
| 360° | 0 | -0 | yes |
The mistake students make
Mixing up which pair to add: sine-cosine products become a sum of two SINES, cosine-cosine products become 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 sin B = ½[sin(A + B) − sin(A − B)]Product to sum: cos A sin Bdetailsproof
- sin A sin B = ½[cos(A − B) − cos(A + B)]Product to sum: sin A sin Bdetailsproof
- cos A cos B = ½[cos(A − B) + cos(A + B)]Product to sum: cos A cos Bdetailsproof
- sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2)Sum to product: sin A + sin Bdetailsproof
Category hub: Product-to-sum identities · all identities: /identities