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

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

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

The proof

One line of substitution, provided the even/odd facts are in hand.

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

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 sin(A − B) = sin A cos B − cos A sin B myself? — the derivation, which is a different question from the proof.
• Sine 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