Double-angle identities · derivation ·proof
Where cos 2x = 1 − 2 sin²x comes from
Solve for sin². Rearranging gives the single most-used formula in integral trigonometry.
cos 2x = 1 − 2 sin²x
Building it step by step
- cos 2x = 1 − 2sin²xStart.
- 2sin²x = 1 − cos 2xMove the sine term to the left.
- sin²x = (1 − cos 2x)/2Power-reducing formula. · Power-reducing formula for sin²
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- sin²x = (1 − cos 2x)/2 — Power-reducing formula for sin²
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 sin²x = (1 − cos 2x)/2 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 3) page.