Reciprocal and quotient identities · trigonometric identity
cot θ = cos θ / sin θ — Quotient identity (cotangent)
When to use it
Same idea as tan = sin/cos with the roles swapped: use it when cot θ and csc θ appear together, because both then share the denominator sin θ.
Why it is true
Cotangent is adj/opp, and dividing cos θ by sin θ cancels the hypotenuse and leaves exactly adj/opp.
The full line-by-line version is on the proof page for quotient identity (cotangent); the “how would I find this myself” version is in the derivation.
Where it comes from
Flip tangent. cot θ is 1/tan θ; flipping sin/cos gives cos/sin.
- cot θ = 1/tan θReciprocal identity. · Reciprocal identity (cotangent)
- tan θ = sin θ/cos θQuotient identity. · Quotient identity (tangent)
- cot θ = cos θ/sin θReciprocal of a fraction.
Worked examples
- Verify cot θ + csc θ = (1 + cos θ)/sin θBoth sides are the same single fraction once cot and csc are written over sin θ.
- Verify tan θ + cot θ = sec θ csc θConvert to sin and cos, take the common denominator, and the Pythagorean identity closes it.
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 |
| 90° | 0 | 0 | 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 |
| 270° | 0 | 0 | yes |
| 300° | -0.57735 | -0.57735 | yes |
| 330° | -1.732051 | -1.732051 | yes |
The mistake students make
Writing cot θ = sin θ/cos θ — that is tangent. The mnemonic: cot starts with c, so it starts with cos.
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
- tan θ = sin θ / cos θQuotient identity (tangent)detailsproof
- cot θ = 1/tan θReciprocal identity (cotangent)detailsproof
- 1 + cot²θ = csc²θPythagorean identity (cotangent form)detailsproof
Category hub: Reciprocal and quotient identities · all identities: /identities