Product-to-sum identities · derivation ·proof
Where sin A sin B = ½[cos(A − B) − cos(A + B)] comes from
A = B gives sin². Specialising this formula is the usual way to reach the sin² reduction.
sin A sin B = ½[cos(A − B) − cos(A + B)]
Building it step by step
- B = ASpecialise.
- cos 0 = 1, cos 2A staysSimplify.
- sin²A = ½(1 − cos 2A)Power-reducing formula. · Power-reducing formula for sin²
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- sin²x = (1 − cos 2x)/2 — Power-reducing formula for sin²
That list is the argument for learning fewer formulas: most of this catalogue is a short chain away from the unit circle and the sum formulas.
How to recall it under pressure
Re-derive it from the first line rather than searching memory: start from sin²x = (1 − cos 2x)/2 and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the product to sum: sin a sin b page.