Cofunction and even/odd identities · trigonometric identity
tan θ = cot(90° − θ) — Cofunction identity for tangent
When to use it
Tangent and cotangent are the complementary pair. Appears in similar-triangle problems and in heights-of-objects word problems where the angle is measured from the other side.
Why it is true
Follows from the sine and cosine cofunctions: tan θ = sin θ/cos θ = cos(90°−θ)/sin(90°−θ) = cot(90°−θ).
The full line-by-line version is on the proof page for cofunction identity for tangent; the “how would I find this myself” version is in the derivation.
Where it comes from
Build it from the pair you trust. Rather than memorising a third cofunction, derive it by dividing the first two.
- Take sin ↔ cos cofunctionsThose have a geometric proof.
- Dividesin/cos is tan; cos/sin is cot.
- tan ↔ cot at complementary anglesConclusion.
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 |
|---|---|---|---|
| 0° | 0 | 0 | yes |
| 7.5° | 0.131652 | 0.131652 | yes |
| 15° | 0.267949 | 0.267949 | yes |
| 18° | 0.32492 | 0.32492 | yes |
| 22.5° | 0.414214 | 0.414214 | yes |
| 30° | 0.57735 | 0.57735 | yes |
| 37° | 0.753554 | 0.753554 | yes |
| 45° | 1 | 1 | yes |
| 53° | 1.327045 | 1.327045 | yes |
| 60° | 1.732051 | 1.732051 | yes |
| 67.5° | 2.414214 | 2.414214 | yes |
| 75° | 3.732051 | 3.732051 | yes |
| 120° | -1.732051 | -1.732051 | yes |
| 135° | -1 | -1 | yes |
| 150° | -0.57735 | -0.57735 | yes |
| 180° | -0 | -0 | yes |
| 210° | 0.57735 | 0.57735 | yes |
| 240° | 1.732051 | 1.732051 | yes |
| 300° | -1.732051 | -1.732051 | yes |
| 330° | -0.57735 | -0.57735 | yes |
| 360° | -0 | -0 | yes |
The mistake students make
Using it outside its domain: cot(90° − θ) is undefined when θ = 0° and tan θ is undefined at θ = 90° — the two sides fail at complementary points, they do not disagree.
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 cofunction and even/odd identities.
Related identities
- sin θ = cos(90° − θ)Cofunction identity for sinedetailsproof
- cos θ = sin(90° − θ)Cofunction identity for cosinedetailsproof
- tan θ = sin θ / cos θQuotient identity (tangent)detailsproof
Category hub: Cofunction and even/odd identities · all identities: /identities