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

Proof: tan(x/2) = (1 − cos x)/sin x

Half-angle identity for tangent (form 1) — proved by half-angle sine over half-angle cosine. Every line below says which rule it uses, so nothing has to be taken on faith.

The proof

Divide sin(x/2) by cos(x/2) and clear the roots.

1. tan(x/2) = sin(x/2)/cos(x/2)
   Quotient identity. · Quotient identity (tangent)
2. multiply numerator and denominator by 2cos(x/2)
   Chooses which double angle appears.
3. = 2 sin(x/2)cos(x/2) / (2cos²(x/2))
   Rearrange.
4. = sin x/(1 + cos x)
   Double-angle formulas. · Double-angle identity for sine, Double-angle identity for cosine (form 2)
5. = (1 − cos x)/sin x
   Multiply top and bottom by (1 − cos x) and use 1 − cos²x = sin²x. · Pythagorean identity

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) = (1 − cos x)/sin x myself? — the derivation, which is a different question from the proof.
• Half-angle identity for tangent (form 1): 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