Reciprocal and quotient identities · derivation ·proof

Where sec θ = 1/cos θ comes from

Flip and name. Secant is 'the reciprocal of cosine' wearing a different name; the derivation is the flip.

Building it step by step

1. cos θ = adj/hyp
   Right-triangle ratio.
2. 1/cos θ = hyp/adj
   Algebra.
3. hyp/adj is sec θ
   Naming.

How to recall it under pressure

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