Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-08-02
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • 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.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Double-angle and quadratic power-reduction identities

Statement

For every real x, sin⁡(2x)=2sin⁡xcos⁡x,cos⁡(2x)=cos⁡2x−sin⁡2x=2cos⁡2x−1=1−2sin⁡2x, and sin⁡2x=(1−cos⁡2x)/2,cos⁡2x=(1+cos⁡2x)/2. When defined, tan⁡(2x)=2tan⁡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 x.

Proof

technique · direct
1.1

Put u=v=x in the addition formulas.

given
1.2

Use 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 · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources