Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Double-angle and quadratic power-reduction identities

Statement

For every real xx, sin⁡(2x)=2sin⁡xcos⁡x,cos⁡(2x)=cos⁡2x−sin⁡2x=2cos⁡2x−1=1−2sin⁡2x,\sin(2x)=2\sin x\cos x,\quad \cos(2x)=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x, and sin⁡2x=(1−cos⁡2x)/2,cos⁡2x=(1+cos⁡2x)/2.\sin^2x=(1-\cos2x)/2,\qquad \cos^2x=(1+\cos2x)/2. When defined, tan⁡(2x)=2tan⁡x/(1−tan⁡2x)\tan(2x)=2\tan x/(1-\tan^2x). The conventions and prerequisite facts used below are recorded in The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine, Tangent, cotangent, secant, and cosecant on their exact natural domains.

Facts & Assumptions

Given: A real xx.

Proof

technique · direct
1.1

Put u=v=xu=v=x in the addition formulas.

given
1.2

Use sin⁡2x+cos⁡2x=1\sin^2x+\cos^2x=1 to rewrite the cosine identity and solve both resulting equalities for the squares.

algebra
2.1

Divide the double-angle sine identity by the double-angle cosine identity only when both quotient expressions are defined.

algebra∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 32 results over 18 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources