Sum and difference identities · derivation ·proof

Where tan(A + B) = (tan A + tan B) / (1 − tan A tan B) comes from

Force tangents to appear. The trick is the division by cos A cos B — it is what turns mixed sine/cosine products into tangents.

Building it step by step

1. Every term should become tan
   Goal.
2. tan = sin/cos needs a cos in the denominator
   So divide by the product cos A cos B.
3. (sin A cos B)/(cos A cos B) = tan A
   Cancellation, term by term.

How to recall it under pressure

Re-derive it from the first line rather than searching memory: start from tan(a + b) and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the tangent of a sum page.

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