Double-angle identities · derivation ·proof
Where cos 2x = 2 cos²x − 1 comes from
Solve for cos². This is the standard route to the power-reducing formula for cosine.
cos 2x = 2 cos²x − 1
Building it step by step
- cos 2x = 2cos²x − 1Start.
- cos 2x + 1 = 2cos²xAdd 1 to both sides.
- cos²x = (1 + cos 2x)/2Divide by 2 — the power-reducing form. · Power-reducing formula for cos²
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- cos²x = (1 + cos 2x)/2 — Power-reducing formula for cos²
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 cos²x = (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 2) page.