Reciprocal and quotient identities · derivation ·proof

Where csc θ = 1/sin θ 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.

Building it step by step

1. sin θ = opp/hyp
   Right-triangle ratio.
2. flip → hyp/opp
   Reciprocal of the ratio.
3. hyp/opp is named csc θ
   Naming convention: csc goes with sin, sec with cos, cot with tan.

How to recall it under pressure

Re-derive it from the first line rather than searching memory: start from csc(x) and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the reciprocal identity (cosecant) page.

sampled at 800 random anglesSource: OpenStax Precalculus, Ch. 7 · Paul's Online Math Notes, Trig Cheat Sheet · revised 2026-09-26 ·how we check ·accuracy policy ·report an error