Half-angle identities · derivation ·proof
Where tan(x/2) = (1 − cos x)/sin x comes from
Rationalise to switch forms. The two tangent half-angle forms are the same fraction with the conjugate multiplied in — you never need to memorise both.
tan(x/2) = (1 − cos x)/sin x
Building it step by step
- Start with sin x/(1 + cos x)One form.
- Multiply by (1 − cos x)/(1 − cos x)Rationalising factor.
- Bottom: 1 − cos²x = sin²xPythagorean identity. · Pythagorean identity
- Cancel one sin x → (1 − cos x)/sin xThe other form. · Half-angle identity for tangent (form 2)
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- sin²θ + cos²θ = 1 — Pythagorean identity
- tan(x/2) = sin x/(1 + cos x) — Half-angle identity for tangent (form 2)
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 sin²θ + cos²θ = 1 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 1) page.