Pythagorean identities · proof ·what it is and when to use it

Proof: 1 + tan²θ = sec²θ

Pythagorean identity (tangent form) — proved by divide the pythagorean identity. Every line below says which rule it uses, so nothing has to be taken on faith.

The proof

Start from the one identity everybody knows and divide every term by cos²θ.

1. sin²θ + cos²θ = 1
   Pythagorean identity. · Pythagorean identity
2. sin²θ/cos²θ + cos²θ/cos²θ = 1/cos²θ
   Divide all three terms by cos²θ (allowed when cos θ ≠ 0).
3. tan²θ + 1 = sec²θ
   Quotient identity tan = sin/cos and reciprocal identity sec = 1/cos. · Quotient identity (tangent), Reciprocal identity (secant)

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 1 + tan²θ = sec²θ myself? — the derivation, which is a different question from the proof.
• Pythagorean identity (tangent form): when to use it — the practical side.
• All pythagorean 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