Identity library · 5 formulas
Double-angle identities
Set b = a in the sum formulas. Cosine comes out in three equivalent shapes — cos²x − sin²x, 2cos²x − 1, 1 − 2sin²x — and picking the right one is half the work: the last two are what let you integrate cos²x or sin²x without a reduction formula.
Every double-angle identities on this site
| Formula | Name | Used for |
|---|---|---|
| sin 2x = 2 sin x cos x | Double-angle identity for sine | simplify, solve, differentiate |
| cos 2x = cos²x − sin²x | Double-angle identity for cosine (form 1) | simplify, verify |
| cos 2x = 2 cos²x − 1 | Double-angle identity for cosine (form 2) | simplify, integrate |
| cos 2x = 1 − 2 sin²x | Double-angle identity for cosine (form 3) | simplify, integrate |
| tan 2x = 2 tan x / (1 − tan²x) | Double-angle identity for tangent | solve, simplify |
Double-angle identities (5)
sin 2x, the three forms of cos 2x, and tan 2x.
| Identity | Name | proof |
|---|---|---|
| sin 2x = 2 sin x cos x | Double-angle identity for sine | proof |
| cos 2x = cos²x − sin²x | Double-angle identity for cosine (form 1) | proof |
| cos 2x = 2 cos²x − 1 | Double-angle identity for cosine (form 2) | proof |
| cos 2x = 1 − 2 sin²x | Double-angle identity for cosine (form 3) | proof |
| tan 2x = 2 tan x / (1 − tan²x) | Double-angle identity for tangent | 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.