Identity library · 4 formulas
Power-reducing identities
Rearrangements of the double-angle formulas whose job is to lower a power. Integration almost always wants them: sin²x = (1 − cos 2x)/2 turns a squared trig term into something a first-year student can integrate in one line.
Every power-reducing identities on this site
| Formula | Name | Used for |
|---|---|---|
| sin²x = (1 − cos 2x)/2 | Power-reducing formula for sin² | integrate, simplify |
| cos²x = (1 + cos 2x)/2 | Power-reducing formula for cos² | integrate, simplify |
| sin³x = (3 sin x − sin 3x)/4 | Reduction formula for sin³ | integrate, simplify |
| cos³x = (3 cos x + cos 3x)/4 | Reduction formula for cos³ | integrate, simplify |
Power-reducing identities (4)
sin², cos², sin³, cos⁴ rewritten with no powers above one.
| Identity | Name | proof |
|---|---|---|
| sin²x = (1 − cos 2x)/2 | Power-reducing formula for sin² | proof |
| cos²x = (1 + cos 2x)/2 | Power-reducing formula for cos² | proof |
| sin³x = (3 sin x − sin 3x)/4 | Reduction formula for sin³ | proof |
| cos³x = (3 cos x + cos 3x)/4 | Reduction formula for cos³ | proof |
Doing something with them
- Which identity should I use? — describe an expression, get the rule.
- Verify — check a claim, see a counterexample if it is false.
- Printable cheat sheet — this family pre-selected.