Sum-to-product identities · proof ·what it is and when to use it

Proof: cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2)

Sum to product: cos A + cos B — proved by add the cosine expansions. Every line below says which rule it uses, so nothing has to be taken on faith.

The proof

u = (A + B)/2, v = (A − B)/2 again.

1. cos A + cos B = cos(u + v) + cos(u − v)
   Substitution.
2. = (cos u cos v − sin u sin v) + (cos u cos v + sin u sin v)
   Cosine sum and difference. · Cosine of a sum, Cosine of a difference
3. = 2 cos u cos v
   Sine products cancel.

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 + cos B = 2 cos((A + B)/2) cos((A − B)/2) myself? — the derivation, which is a different question from the proof.
• Sum to product: cos A + cos B: when to use it — the practical side.
• All sum-to-product 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