Product-to-sum identities · derivation ·proof
Where sin A cos B = ½[sin(A + B) + sin(A − B)] comes from
Never memorise — re-derive in four lines. There are four product-to-sum formulas and two directions. Memorising eight lines is worse than knowing the trick: expand the sum and the difference, then add or subtract.
sin A cos B = ½[sin(A + B) + sin(A − B)]
Building it step by step
- Write the sum formula and the difference formula for the function you havesin here.
- Add them if the product is mixed (sin·cos)Mixed products survive addition.
- Subtract them if the product matches (sin·sin or cos·cos)Same-function products survive subtraction.
- Divide by 2Each surviving term appears twice.
How to recall it under pressure
Re-derive it from the first line rather than searching memory: start from sin(a)*cos(b) 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 cos b page.