Reciprocal and quotient identities · derivation ·proof
Where cot θ = cos θ / sin θ comes from
Flip tangent. cot θ is 1/tan θ; flipping sin/cos gives cos/sin.
cot θ = cos θ / sin θ
Building it step by step
- 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.
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- cot θ = 1/tan θ — Reciprocal identity (cotangent)
- tan θ = sin θ / cos θ — Quotient identity (tangent)
That list is the argument for learning fewer formulas: most of this catalogue is a short chain away from the unit circle and the sum formulas.
How to recall it under pressure
Re-derive it from the first line rather than searching memory: start from cot θ = 1/tan θ and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the quotient identity (cotangent) page.