Reciprocal and quotient identities · proof ·what it is and when to use it

Proof: tan θ = sin θ / cos θ

Quotient identity (tangent) — proved by slope of the radius. Every line below says which rule it uses, so nothing has to be taken on faith.

The proof

Two ways to see it: the triangle ratio, or the slope on the unit circle.

1. sin θ = opp/hyp, cos θ = adj/hyp
   Right-triangle definitions.
2. sin θ/cos θ = (opp/hyp)·(hyp/adj) = opp/adj
   Divide by multiplying by the reciprocal; hyp cancels.
3. opp/adj = tan θ
   TOA — so tan θ = sin θ/cos θ.

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 tan θ = sin θ / cos θ myself? — the derivation, which is a different question from the proof.
• Quotient identity (tangent): when to use it — the practical side.
• All reciprocal and quotient 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