Product-to-sum identities · derivation ·proof

Where cos A sin B = ½[sin(A + B) − sin(A − B)] comes from

Track which term cancels. Addition keeps the terms that appear with the same sign in both expansions; subtraction keeps the ones that appear with opposite signs.

Building it step by step

1. Expand both
   Standard move.
2. Want cos A sin B? It has opposite signs
   So subtract.
3. ½[sin(A + B) − sin(A − B)]
   Result.

How to recall it under pressure

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