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

Proof: sin θ = cos(90° − θ)

Cofunction identity for sine — proved by the two acute angles of a right triangle. Every line below says which rule it uses, so nothing has to be taken on faith.

The proof

One triangle, two labels for the same ratio.

1. A + B = 90°
   Acute angles of a right triangle are complementary.
2. sin A = opp(A)/hyp = adj(B)/hyp
   The side facing A is the side beside B.
3. adj(B)/hyp = cos B
   Definition of cosine at B.
4. sin A = cos(90° − A)
   Substitute B = 90° − A.

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