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

Proof: tan(A − B) = (tan A − tan B) / (1 + tan A tan B)

Tangent 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

Same one-line move as the sine and cosine difference formulas.

1. tan(A + B) = (tan A + tan B)/(1 − tan A tan B)
   Sum version. · Tangent of a sum
2. B → −B, tan(−B) = −tan B
   Tangent is odd. · Tangent is odd
3. tan(A − B) = (tan A − tan B)/(1 + tan A tan B)
   Substitute and simplify signs.

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 − B) = (tan A − tan B) / (1 + tan A tan B) myself? — the derivation, which is a different question from the proof.
• Tangent of a difference: when to use it — the practical side.
• All sum and difference identities · proof index

sampled at 800 random angles · worst relative difference 1.2e-12Source: OpenStax Precalculus, Ch. 7.2–7.3 · Paul's Online Math Notes · revised 2026-09-26 ·how we check ·accuracy policy ·report an error