Sum-to-product identities · derivation ·proof
Where cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2) comes from
Sanity-check with A = B. When A = B the left side is 0 and the right side is 0 because sin 0 = 0 — consistent. Then one numeric test fixes the overall sign.
cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2)
Building it step by step
- A = B: both sides 0Necessary condition ✓.
- A = 0, B = π: left = 1 − (−1) = 2Concrete test.
- right = −2 sin(π/2) sin(−π/2) = +2Matches only with the leading minus.
How to recall it under pressure
Re-derive it from the first line rather than searching memory: start from cos(a) - cos(b) and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the sum to product: cos a − cos b page.