Double-angle identities · derivation ·proof
Where cos 2x = cos²x − sin²x comes from
From the symmetric form to the other two. The other two cosine forms are this one plus the Pythagorean identity, used to eliminate whichever square you do not want.
cos 2x = cos²x − sin²x
Building it step by step
- cos²x = 1 − sin²xPythagorean identity. · Pythagorean identity
- Substitute into cos²x − sin²xEliminate cosine.
- cos 2x = 1 − 2 sin²xThird form. · Double-angle identity for cosine (form 3)
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- sin²θ + cos²θ = 1 — Pythagorean identity
- cos 2x = 1 − 2 sin²x — Double-angle identity for cosine (form 3)
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 sin²θ + cos²θ = 1 and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the double-angle identity for cosine (form 1) page.