Half-angle identities · derivation ·proof
Where tan(x/2) = csc x − cot x comes from
Read the names off the pieces. If you can derive form 1, this one is only naming: 1/sin is csc, cos/sin is cot.
tan(x/2) = csc x − cot x
Building it step by step
- 1/sin x is csc xReciprocal. · Reciprocal identity (cosecant)
- cos x/sin x is cot xQuotient. · Quotient identity (cotangent)
- Difference of the twoConclusion.
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- csc θ = 1/sin θ — Reciprocal identity (cosecant)
- cot θ = cos θ / sin θ — Quotient identity (cotangent)
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 csc θ = 1/sin θ and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the half-angle identity for tangent (form 3) page.