Cofunction and even/odd identities · proof ·what it is and when to use it

Proof: cos(−θ) = cos θ

Cosine is even — proved by same reflection. Every line below says which rule it uses, so nothing has to be taken on faith.

The proof

Read the x-coordinate off the reflected point.

1. P(θ) = (cos θ, sin θ)
   Definition.
2. P(−θ) = (cos θ, −sin θ)
   Reflection across the x-axis.
3. cos(−θ) = cos θ
   The x-coordinate is unchanged — cosine is even.

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(−θ) = cos θ myself? — the derivation, which is a different question from the proof.
• Cosine is even: when to use it — the practical side.
• All cofunction and even/odd identities · proof index

sampled at 800 random anglesSource: OpenStax Precalculus, Ch. 7 · Paul's Online Math Notes, Trig Cheat Sheet · revised 2026-09-26 ·how we check ·accuracy policy ·report an error