Tool · runs in your browser
Product to sum
Products of trig functions do not integrate and rarely cancel. These four rules turn them into sums, where each term can be handled on its own.
The rules this converter uses
Nothing is invented in the tool: these are 4 identities from the product to sum family, and the converter only applies one when the expression actually matches it.
| Rule | Name |
|---|---|
| sin A cos B = ½[sin(A + B) + sin(A − B)] | Product to sum: sin A cos B |
| cos A sin B = ½[sin(A + B) − sin(A − B)] | Product to sum: cos A sin B |
| cos A cos B = ½[cos(A − B) + cos(A + B)] | Product to sum: cos A cos B |
| sin A sin B = ½[cos(A − B) − cos(A + B)] | Product to sum: sin A sin B |
How the result is checked
After converting, the tool evaluates the original and the converted expression at 12 random angles. It only prints a conversion that agrees at every one of them — a transformation that fails that check is dropped rather than shown.
- The angle in the answer is the SUM and the DIFFERENCE of the two angles, each halved by the ½ factor.
- For a squared term (sin x cos x with the same angle), the double-angle identity is usually shorter.
Other directions
- Sum to product calculator
- Power reducing calculator (sin², cos², sin³, cos³)
- Degrees to radians
- Radians to degrees
Want the rule identified for you instead of picking a direction?Which identity should I use?