Tool · runs in your browser
Power reducing
A power above one has no direct integral. These formulas replace the square or cube with a sum of first powers of x, 2x, 3x.
The rules this converter uses
Nothing is invented in the tool: these are 4 identities from the power reducing family, and the converter only applies one when the expression actually matches it.
| Rule | Name |
|---|---|
| sin²x = (1 − cos 2x)/2 | Power-reducing formula for sin² |
| cos²x = (1 + cos 2x)/2 | Power-reducing formula for cos² |
| sin³x = (3 sin x − sin 3x)/4 | Reduction formula for sin³ |
| cos³x = (3 cos x + cos 3x)/4 | Reduction formula for cos³ |
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.
- Squared terms introduce cos 2x; cubic terms introduce sin 3x or cos 3x.
- Check the result: adding the sin² and cos² reductions must give 1.
Other directions
Want the rule identified for you instead of picking a direction?Which identity should I use?