Sum-to-product identities · derivation ·proof

Where sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2) comes from

Keep the half-difference where the change is. The term with (A − B)/2 is the 'small' one; that is the factor that vanishes when A = B, which is a good sanity check.

Building it step by step

1. Check A = B: the right side must be 0
   sin 0 = 0 kills the product ✓.
2. So sin((A − B)/2) has to be a factor
   Structural necessity.
3. The other factor is the cosine of the average
   Symmetry of the two angles.

How to recall it under pressure

Re-derive it from the first line rather than searching memory: start from sin(a) - sin(b) and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the sum to product: sin a − sin b page.

sampled at 800 random anglesSource: OpenStax Precalculus, Ch. 7.2–7.3 · Paul's Online Math Notes · revised 2026-09-26 ·how we check ·accuracy policy ·report an error