Reciprocal and quotient identities · derivation ·proof
Where cot θ = 1/tan θ comes from
Two roads to the same ratio. Cotangent can be reached from tangent's reciprocal or from cos/sin; both routes agree, which is exactly what the quotient identity says.
cot θ = 1/tan θ
Building it step by step
- cot θ = 1/tan θReciprocal.
- tan θ = sin θ/cos θQuotient identity. · Quotient identity (tangent)
- cot θ = cos θ/sin θSubstitute and flip.
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- 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 tan θ = sin θ / cos θ and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the reciprocal identity (cotangent) page.