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

Proof: tan(x/2) = csc x − cot x

Half-angle identity for tangent (form 3) — proved by split the fraction. Every line below says which rule it uses, so nothing has to be taken on faith.

The proof

Three lines from form 1.

1. tan(x/2) = (1 − cos x)/sin x
   Half-angle form 1. · Half-angle identity for tangent (form 1)
2. = 1/sin x − cos x/sin x
   Split the numerator.
3. = csc x − cot x
   Reciprocal and quotient identities. · Reciprocal identity (cosecant), Quotient identity (cotangent)

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) = csc x − cot x myself? — the derivation, which is a different question from the proof.
• Half-angle identity for tangent (form 3): 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