Sum and difference identities · derivation ·proof
Where cos(A + B) = cos A cos B − sin A sin B comes from
Derive the double-angle formula from it. Setting B = A is the standard route to cos 2A, and from there to the three cosine double-angle forms.
cos(A + B) = cos A cos B − sin A sin B
Building it step by step
- cos(A + B) = cos A cos B − sin A sin BStart. · Cosine of a sum
- B = ASpecialise.
- cos 2A = cos²A − sin²AMultiply matching functions. · Double-angle identity for cosine (form 1)
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- cos(A + B) = cos A cos B − sin A sin B — Cosine of a sum
- cos 2x = cos²x − sin²x — Double-angle identity for cosine (form 1)
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 cos(A + B) = cos A cos B − sin A sin B and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the cosine of a sum page.