Pythagorean identities · derivation ·proof
Where 1 + cot²θ = csc²θ comes from
Force the cotangent to appear. Cotangent is cos over sin, so dividing by sin²θ is the only division that produces it.
1 + cot²θ = csc²θ
Building it step by step
- cot²θ = cos²θ/sin²θQuotient identity. · Quotient identity (cotangent)
- 1/sin²θ = csc²θReciprocal identity. · Reciprocal identity (cosecant)
- 1 + cot²θ = csc²θSubstitute both into the divided Pythagorean identity.
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- cot θ = cos θ / sin θ — Quotient identity (cotangent)
- csc θ = 1/sin θ — Reciprocal identity (cosecant)
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 = cos θ / sin θ and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the pythagorean identity (cotangent form) page.