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

Proof: tan A + tan B = sin(A + B) / (cos A cos B)

Tangent sum as a single fraction — proved by common denominator. Every line below says which rule it uses, so nothing has to be taken on faith.

The proof

Two fractions, one denominator.

1. tan A + tan B = sin A/cos A + sin B/cos B
   Quotient identity. · Quotient identity (tangent)
2. = (sin A cos B + cos A sin B)/(cos A cos B)
   Common denominator.
3. = sin(A + B)/(cos A cos B)
   Numerator is the sine sum. · Sine of a sum

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 tan A + tan B = sin(A + B) / (cos A cos B) myself? — the derivation, which is a different question from the proof.
• Tangent sum as a single fraction: 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