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

Proof: cos θ = sin(90° − θ)

Cofunction identity for cosine — proved by same triangle, other angle. Every line below says which rule it uses, so nothing has to be taken on faith.

The proof

Mirror of the sine proof.

1. A + B = 90°
   Complementary acute angles.
2. cos A = adj(A)/hyp = opp(B)/hyp
   Adjacent to A is opposite to B.
3. opp(B)/hyp = sin B
   Definition of sine at B.
4. cos A = sin(90° − A)
   Substitute.

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 = sin(90° − θ) myself? — the derivation, which is a different question from the proof.
• Cofunction identity for cosine: 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