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

Proof: cos 2x = cos²x − sin²x

Double-angle identity for cosine (form 1) — proved by set b = a in the cosine sum. Every line below says which rule it uses, so nothing has to be taken on faith.

The proof

Direct specialisation.

1. cos(A + B) = cos A cos B − sin A sin B
   Cosine of a sum. · Cosine of a sum
2. A = B = x
   Specialise.
3. cos 2x = cos²x − sin²x
   Multiply matching functions.

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 cos 2x = cos²x − sin²x myself? — the derivation, which is a different question from the proof.
• Double-angle identity for cosine (form 1): when to use it — the practical side.
• All double-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