Sum and difference identities · derivation ·proof
Where tan A + tan B = sin(A + B) / (cos A cos B) comes from
Ask what the numerator wants to be. If a verification problem hands you tan A + tan B and asks for a single sine over a product of cosines, the middle step is always the common denominator.
tan A + tan B = sin(A + B) / (cos A cos B)
Building it step by step
- Target numerator: sin(A + B)What the exercise asks for.
- Expand the target: sin A cos B + cos A sin BThat is exactly what a common denominator produces. · Sine of a sum
- So the route is: quotient, then common denominatorConclusion.
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- sin(A + B) = sin A cos B + cos A sin B — Sine of a sum
That list is the argument for learning fewer formulas: most of this catalogue is a short chain away from the unit circle and the sum formulas.
How to recall it under pressure
Re-derive it from the first line rather than searching memory: start from sin(A + B) = sin A cos B + cos A sin B and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the tangent sum as a single fraction page.