Power-reducing identities · derivation ·proof

Where cos²x = (1 + cos 2x)/2 comes from

Check against the Pythagorean identity. sin² + cos² from these two formulas must give 1 — the cos 2x terms cancel.

Building it step by step

1. (1 − cos 2x)/2 + (1 + cos 2x)/2
   Add the two reduction formulas. · Power-reducing formula for sin², Power-reducing formula for cos²
2. = 2/2 = 1
   Cosine terms cancel — consistent with sin² + cos² = 1. · Pythagorean identity

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
• cos²x = (1 + cos 2x)/2 — Power-reducing formula for cos²
• sin²θ + cos²θ = 1 — Pythagorean identity

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 power-reducing formula for cos² page.

sampled at 800 random anglesSource: OpenStax Precalculus, Ch. 7.3 · Paul's Online Math Notes · revised 2026-09-26 ·how we check ·accuracy policy ·report an error