Power-reducing identities · proof ·what it is and when to use it

Proof: cos²x = (1 + cos 2x)/2

Power-reducing formula for cos² — proved by rearrange double-angle form 2. Every line below says which rule it uses, so nothing has to be taken on faith.

The proof

Same algebra, other sign.

1. cos 2x = 2cos²x − 1
   Double-angle form 2. · Double-angle identity for cosine (form 2)
2. cos 2x + 1 = 2cos²x
   Add 1.
3. cos²x = (1 + cos 2x)/2
   Divide by 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 cos²x = (1 + cos 2x)/2 myself? — the derivation, which is a different question from the proof.
• Power-reducing formula for cos²: when to use it — the practical side.
• All power-reducing 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