Half-angle identities · proof ·what it is and when to use it

Proof: tan(x/2) = sin x/(1 + cos x)

Half-angle identity for tangent (form 2) — proved by stop halfway. Every line below says which rule it uses, so nothing has to be taken on faith.

The proof

The first four lines of the form-1 proof, without the last step.

1. tan(x/2) = sin(x/2)/cos(x/2)
   Quotient identity.
2. multiply by 2cos(x/2)/2cos(x/2)
   Set up the double angles.
3. = 2sin(x/2)cos(x/2)/(2cos²(x/2))
   Algebra.
4. = sin x/(1 + cos x)
   Double-angle formulas. · Double-angle identity for sine, Double-angle identity for cosine (form 2)

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(x/2) = sin x/(1 + cos x) myself? — the derivation, which is a different question from the proof.
• Half-angle identity for tangent (form 2): when to use it — the practical side.
• All half-angle identities · proof index

sampled at 800 random anglesSource: OpenStax Precalculus, Ch. 7.3 · Paul's Online Math Notes · revised 2026-09-26 ·how we check ·accuracy policy ·report an error