Power-reducing identities · derivation ·proof
Where sin²x = (1 − cos 2x)/2 comes from
Powers are eliminated by doubling the angle. General rule of thumb in this family: a squared trig function becomes a constant plus a cosine of DOUBLE the angle. Cube powers become a difference of sines of x and 3x.
sin²x = (1 − cos 2x)/2
Building it step by step
- Need to lose sin²?Set-up.
- Find the double-angle formula containing sin²
- Solve for sin²Algebra.
What this derivation depends on
Every line above is one of these — nothing else is assumed:
- 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 cos 2x = 1 − 2 sin²x and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the power-reducing formula for sin² page.