Pythagorean identities · derivation ·proof
Where 1 + tan²θ = sec²θ comes from
Same division, aimed forward. If you have forgotten the formula, re-derive it in ten seconds: you want a relation involving tan, so force tan = sin/cos to appear by dividing by cos²θ.
1 + tan²θ = sec²θ
Building it step by step
- Want tan²θ? Divide by cos²θ.tan²θ = sin²θ/cos²θ, so the division creates exactly the term you need.
- 1/cos²θ is sec²θReciprocal identity. · Reciprocal identity (secant)
- 1 + tan²θ = sec²θCollect the three pieces.
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- sec θ = 1/cos θ — Reciprocal identity (secant)
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 sec θ = 1/cos θ and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the pythagorean identity (tangent form) page.