Reciprocal and quotient identities · proof ·what it is and when to use it
Proof: cot θ = cos θ / sin θ
Quotient identity (cotangent) — proved by triangle ratios. Every line below says which rule it uses, so nothing has to be taken on faith.
cot θ = cos θ / sin θ
The proof
Mirror image of the tangent proof.
- cos θ/sin θ = (adj/hyp)/(opp/hyp)Definitions.
- = adj/oppThe hypotenuse cancels.
- adj/opp = cot θDefinition of cotangent.
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).
| Angle | Left side | Right side | Agree |
|---|---|---|---|
| 7.5° | 7.595754 | 7.595754 | yes |
| 15° | 3.732051 | 3.732051 | yes |
| 18° | 3.077684 | 3.077684 | yes |
| 22.5° | 2.414214 | 2.414214 | yes |
| 30° | 1.732051 | 1.732051 | yes |
| 37° | 1.327045 | 1.327045 | yes |
Related
- How would I find cot θ = cos θ / sin θ myself? — the derivation, which is a different question from the proof.
- Quotient identity (cotangent): when to use it — the practical side.
- All reciprocal and quotient identities · proof index