Sum and difference identities · proof ·what it is and when to use it

Proof: cos(A − B) = cos A cos B + sin A sin B

Cosine of a difference — proved by substitute −b. Every line below says which rule it uses, so nothing has to be taken on faith.

The proof

Mirror of the sine difference proof.

1. cos(A + B) = cos A cos B − sin A sin B
   Cosine of a sum. · Cosine of a sum
2. B → −B
   A − B = A + (−B).
3. = cos A cos B − sin A sin(−B)
   cos(−B) = cos B. · Cosine is even
4. = cos A cos B + sin A sin B
   sin(−B) = −sin B. · Sine is odd

Where the proof stops applying

Both sides are undefined at the same angles — where a denominator in the proof reaches zero (for instance cos x = 0 in a tangent or secant form). Elsewhere the argument above holds for every real angle.

The same claim, checked numerically

A proof is not the same thing as a check, and this page shows both: the reasoning above, and the first few angles fed to the same engine behind the verifier. If they ever disagreed, the data would be broken — the build would fail before publishing (accuracy policy).

Related

• How would I find cos(A − B) = cos A cos B + sin A sin B myself? — the derivation, which is a different question from the proof.
• Cosine of a difference: when to use it — the practical side.
• All sum and difference identities · proof index

sampled at 800 random anglesSource: OpenStax Precalculus, Ch. 7.2–7.3 · Paul's Online Math Notes · revised 2026-09-26 ·how we check ·accuracy policy ·report an error