Reciprocal and quotient identities · trigonometric identity
cot θ = 1/tan θ — Reciprocal identity (cotangent)
When to use it
cot θ becomes 1/tan θ, and in practice it usually becomes cos θ/sin θ instead — the quotient identity is more useful because it removes the fraction-in-fraction.
Why it is true
Cotangent is the reciprocal of tangent: tan θ = opp/adj, so adj/opp = cot θ.
The full line-by-line version is on the proof page for reciprocal identity (cotangent); the “how would I find this myself” version is in the derivation.
Where it comes from
Two roads to the same ratio. Cotangent can be reached from tangent's reciprocal or from cos/sin; both routes agree, which is exactly what the quotient identity says.
- cot θ = 1/tan θReciprocal.
- tan θ = sin θ/cos θQuotient identity. · Quotient identity (tangent)
- cot θ = cos θ/sin θSubstitute and flip.
Worked examples
No worked example is written for this identity yet —the examples index lists what is covered, and the pattern below shows the identity working at real angles.
Checked at these angles
Two sides of the identity evaluated with the same engine that runs the verifier. A single line disagreeing would break the build.
| 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 |
| 45° | 1 | 1 | yes |
| 53° | 0.753554 | 0.753554 | yes |
| 60° | 0.57735 | 0.57735 | yes |
| 67.5° | 0.414214 | 0.414214 | yes |
| 75° | 0.267949 | 0.267949 | yes |
| 120° | -0.57735 | -0.57735 | yes |
| 135° | -1 | -1 | yes |
| 150° | -1.732051 | -1.732051 | yes |
| 210° | 1.732051 | 1.732051 | yes |
| 240° | 0.57735 | 0.57735 | yes |
| 300° | -0.57735 | -0.57735 | yes |
| 330° | -1.732051 | -1.732051 | yes |
The mistake students make
Forgetting that cot θ is undefined where sin θ = 0 (θ = 0°, 180°) as well as where tan θ = 0. Both descriptions exclude the same points.
Try it
- Verify this identity — both sides are pre-filled; change one character and watch the counterexample appear.
- Ask “which identity should I use?” about the left-hand side — this identity is what the chooser should rank first.
- Cheat sheet entry for reciprocal and quotient identities.
Related identities
- cot θ = cos θ / sin θQuotient identity (cotangent)detailsproof
- csc θ = 1/sin θReciprocal identity (cosecant)detailsproof
- sec θ = 1/cos θReciprocal identity (secant)detailsproof
Category hub: Reciprocal and quotient identities · all identities: /identities