Sum and difference identities · derivation ·proof
Where cos(A − B) = cos A cos B + sin A sin B comes from
Distance between two points on the circle. Put two unit-circle points at angles A and B and compute the squared distance between them twice — once with the distance formula, once with the angle difference.
cos(A − B) = cos A cos B + sin A sin B
Building it step by step
- P = (cos A, sin A), Q = (cos B, sin B)Two points on the unit circle.
- PQ² = (cos A − cos B)² + (sin A − sin B)² = 2 − 2(cos A cos B + sin A sin B)Expand; sin² + cos² = 1 twice. · Pythagorean identity
- PQ² = 2 − 2 cos(A − B)Central angle A − B, chord length.
- cos(A − B) = cos A cos B + sin A sin BEquate the two expressions.
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- sin²θ + cos²θ = 1 — Pythagorean identity
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 sin²θ + cos²θ = 1 and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the cosine of a difference page.