Pythagorean identities · trigonometric identity
1 + cot²θ = csc²θ — Pythagorean identity (cotangent form)
When to use it
The cotangent cousin: use it when 1 + cot²θ shows up, or when csc²θ has to become 1 + cot²θ. It is the same move as the tangent version, aimed at the sine side of the triangle.
Why it is true
Divide sin²θ + cos²θ = 1 by sin²θ instead of cos²θ. The three terms become cot²θ, 1 and csc²θ.
The full line-by-line version is on the proof page for pythagorean identity (cotangent form); the “how would I find this myself” version is in the derivation.
Where it comes from
Force the cotangent to appear. Cotangent is cos over sin, so dividing by sin²θ is the only division that produces it.
- cot²θ = cos²θ/sin²θQuotient identity. · Quotient identity (cotangent)
- 1/sin²θ = csc²θReciprocal identity. · Reciprocal identity (cosecant)
- 1 + cot²θ = csc²θSubstitute both into the divided Pythagorean identity.
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° | 58.695481 | 58.695481 | yes |
| 15° | 14.928203 | 14.928203 | yes |
| 18° | 10.472136 | 10.472136 | yes |
| 22.5° | 6.828427 | 6.828427 | yes |
| 30° | 4 | 4 | yes |
| 37° | 2.761048 | 2.761048 | yes |
| 45° | 2 | 2 | yes |
| 53° | 1.567844 | 1.567844 | yes |
| 60° | 1.333333 | 1.333333 | yes |
| 67.5° | 1.171573 | 1.171573 | yes |
| 75° | 1.071797 | 1.071797 | yes |
| 90° | 1 | 1 | yes |
| 120° | 1.333333 | 1.333333 | yes |
| 135° | 2 | 2 | yes |
| 150° | 4 | 4 | yes |
| 210° | 4 | 4 | yes |
| 240° | 1.333333 | 1.333333 | yes |
| 270° | 1 | 1 | yes |
| 300° | 1.333333 | 1.333333 | yes |
| 330° | 4 | 4 | yes |
The mistake students make
Mixing the two Pythagorean cousins: 1 + tan²θ goes with sec²θ, 1 + cot²θ goes with csc²θ. The initial letters match (t→s, c→c) — tangent/secant, cotangent/cosecant.
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 pythagorean identities.
Related identities
- sin²θ + cos²θ = 1Pythagorean identitydetailsproof
- 1 + tan²θ = sec²θPythagorean identity (tangent form)detailsproof
- cot θ = cos θ / sin θQuotient identity (cotangent)detailsproof
- csc θ = 1/sin θReciprocal identity (cosecant)detailsproof
Category hub: Pythagorean identities · all identities: /identities