Sum and difference identities · derivation ·proof
Where sin(A − B) = sin A cos B − cos A sin B comes from
Remember one formula, derive the other. There is no need to memorise the sum and difference versions separately — the sign of the middle term follows from the sign inside the angle.
sin(A − B) = sin A cos B − cos A sin B
Building it step by step
- Sum version has '+' on both sidessin(A + B) = … + …
- Difference version flips the second termReplace B by −B.
- Mnemonic: sine keeps the signsin(A ± B) = … ± … mirrors the sign inside.
How to recall it under pressure
Re-derive it from the first line rather than searching memory: start from sin(a - b) and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the sine of a difference page.