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 sum — proved by rotation on the unit circle. Every line below says which rule it uses, so nothing has to be taken on faith.

The proof

Place a point at angle A + B. It is the point at angle A, rotated further by B. Coordinates rotate linearly, which is exactly what the formula records.

1. P = (cos(A + B), sin(A + B))
   Unit-circle point at angle A + B.
2. Start from Q = (cos A, sin A) and rotate by B
   A + B = A then B.
3. rotation by B: (x, y) → (x cos B − y sin B, x sin B + y cos B)
   Standard rotation matrix, itself from the cofunction/length argument.
4. y-coordinate = sin A cos B + cos A sin B
   Substitute x = cos A, y = sin A into the rotated y.
5. sin(A + B) = sin A cos B + cos A sin B
   The rotated y must equal sin(A + B).

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 sum: 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