Reciprocal and quotient identities · trigonometric identity
csc θ = 1/sin θ — Reciprocal identity (cosecant)
When to use it
Any time csc θ is in your way, the first move is to replace it with 1/sin θ. The reciprocal functions exist for notation, not for algebra — the algebra happens in sin and cos.
Why it is true
Cosecant is defined as the reciprocal of sine: in a right triangle, sin θ = opposite/hypotenuse, so its reciprocal is hypotenuse/opposite, which is named csc θ.
The full line-by-line version is on the proof page for reciprocal identity (cosecant); the “how would I find this myself” version is in the derivation.
Where it comes from
Flip the fraction. If you are ever unsure which function is whose reciprocal, write the sine/cosine/tangent ratio and turn the fraction upside down.
- sin θ = opp/hypRight-triangle ratio.
- flip → hyp/oppReciprocal of the ratio.
- hyp/opp is named csc θNaming convention: csc goes with sin, sec with cos, cot with tan.
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° | 7.661298 | 7.661298 | yes |
| 15° | 3.863703 | 3.863703 | yes |
| 18° | 3.236068 | 3.236068 | yes |
| 22.5° | 2.613126 | 2.613126 | yes |
| 30° | 2 | 2 | yes |
| 37° | 1.66164 | 1.66164 | yes |
| 45° | 1.414214 | 1.414214 | yes |
| 53° | 1.252136 | 1.252136 | yes |
| 60° | 1.154701 | 1.154701 | yes |
| 67.5° | 1.082392 | 1.082392 | yes |
| 75° | 1.035276 | 1.035276 | yes |
| 90° | 1 | 1 | yes |
| 120° | 1.154701 | 1.154701 | yes |
| 135° | 1.414214 | 1.414214 | yes |
| 150° | 2 | 2 | yes |
| 210° | -2 | -2 | yes |
| 240° | -1.154701 | -1.154701 | yes |
| 270° | -1 | -1 | yes |
| 300° | -1.154701 | -1.154701 | yes |
| 330° | -2 | -2 | yes |
The mistake students make
Reading csc θ as 'cosecant times θ' or as c(·)s(·)c(·). It is one function name: csc, the reciprocal of sin, not a product.
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
- sec θ = 1/cos θReciprocal identity (secant)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