Half-angle identities · derivation ·proof
Where tan(x/2) = sin x/(1 + cos x) comes from
Which form is convenient?. Pick by what is already in your expression: a bare sin x on top suggests this form; a 1 − cos x suggests the other. Neither is more correct.
tan(x/2) = sin x/(1 + cos x)
Building it step by step
- Numerator has sin xThen use this form.
- Denominator has 1 − cos x elsewhere in the problemThen use form 1. · Half-angle identity for tangent (form 1)
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- tan(x/2) = (1 − cos x)/sin x — Half-angle identity for tangent (form 1)
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 tan(x/2) = (1 − cos x)/sin x and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the half-angle identity for tangent (form 2) page.