Identity library · 7 formulas
Sum and difference identities
Six formulas that split one angle into two, or fuse two angles into one. They are the source that double-angle, half-angle and power-reducing formulas are all derived from, and they are what you use when an angle like 15° or 75° has to be computed without a calculator.
Every sum and difference identities on this site
| Formula | Name | Used for |
|---|---|---|
| sin(A + B) = sin A cos B + cos A sin B | Sine of a sum | evaluate, expand, verify |
| sin(A − B) = sin A cos B − cos A sin B | Sine of a difference | evaluate, expand, verify |
| cos(A + B) = cos A cos B − sin A sin B | Cosine of a sum | evaluate, expand, verify |
| cos(A − B) = cos A cos B + sin A sin B | Cosine of a difference | evaluate, verify |
| tan(A + B) = (tan A + tan B) / (1 − tan A tan B) | Tangent of a sum | evaluate, verify, solve |
| tan(A − B) = (tan A − tan B) / (1 + tan A tan B) | Tangent of a difference | evaluate, verify |
| tan A + tan B = sin(A + B) / (cos A cos B) | Tangent sum as a single fraction | verify, simplify |
Sum and difference identities (7)
sin(a ± b), cos(a ± b), tan(a ± b) expanded.
| Identity | Name | proof |
|---|---|---|
| sin(A + B) = sin A cos B + cos A sin B | Sine of a sum | proof |
| sin(A − B) = sin A cos B − cos A sin B | Sine of a difference | proof |
| cos(A + B) = cos A cos B − sin A sin B | Cosine of a sum | proof |
| cos(A − B) = cos A cos B + sin A sin B | Cosine of a difference | proof |
| tan(A + B) = (tan A + tan B) / (1 − tan A tan B) | Tangent of a sum | proof |
| tan(A − B) = (tan A − tan B) / (1 + tan A tan B) | Tangent of a difference | proof |
| tan A + tan B = sin(A + B) / (cos A cos B) | Tangent sum as a single fraction | 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.