Identity library · 5 formulas
Half-angle identities
Solve the double-angle formulas for the half angle. The square root is unavoidable, so the sign is a real question: cos(x/2) = +√((1+cos x)/2) holds for x in (−180°, 180°) and fails outside it. Our pages state the condition instead of hiding it.
Every half-angle identities on this site
| Formula | Name | Used for |
|---|---|---|
| sin(x/2) = ±√((1 − cos x)/2) | Half-angle identity for sine | integrate, simplify, evaluate |
| cos(x/2) = ±√((1 + cos x)/2) | Half-angle identity for cosine | integrate, simplify, evaluate |
| tan(x/2) = (1 − cos x)/sin x | Half-angle identity for tangent (form 1) | integrate, simplify |
| tan(x/2) = sin x/(1 + cos x) | Half-angle identity for tangent (form 2) | integrate, simplify |
| tan(x/2) = csc x − cot x | Half-angle identity for tangent (form 3) | integrate, simplify |
Half-angle identities (5)
sin(x/2) and cos(x/2) — with the sign condition spelled out.
| Identity | Name | proof |
|---|---|---|
| sin(x/2) = ±√((1 − cos x)/2) | Half-angle identity for sine | proof |
| cos(x/2) = ±√((1 + cos x)/2) | Half-angle identity for cosine | proof |
| tan(x/2) = (1 − cos x)/sin x | Half-angle identity for tangent (form 1) | proof |
| tan(x/2) = sin x/(1 + cos x) | Half-angle identity for tangent (form 2) | proof |
| tan(x/2) = csc x − cot x | Half-angle identity for tangent (form 3) | 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.