Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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.

Product-to-sum and sum-to-product identities

Statement

For all real u,vu,v, sin⁡usin⁡v=cos⁡(u−v)−cos⁡(u+v)2,cos⁡ucos⁡v=cos⁡(u−v)+cos⁡(u+v)2,\sin u\sin v=\frac{\cos(u-v)-\cos(u+v)}2,\quad \cos u\cos v=\frac{\cos(u-v)+\cos(u+v)}2, sin⁡ucos⁡v=sin⁡(u+v)+sin⁡(u−v)2,\sin u\cos v=\frac{\sin(u+v)+\sin(u-v)}2, and, with a=(u+v)/2a=(u+v)/2, b=(u−v)/2b=(u-v)/2, the reverse sum-to-product identities follow by solving these formulas for the sums. The conventions and prerequisite facts used below are recorded in The addition formulas for sine and cosine, The subtraction formulas for sine and cosine.

Facts & Assumptions

Given: Reals u,vu,v.

Proof

technique · direct
1.1

Add and subtract the addition and subtraction formulas for cosine.

algebra
1.2

Add the addition and subtraction formulas for sine.

algebra
2.1

Substitute u=a+bu=a+b, v=a−bv=a-b to obtain every reverse identity.

algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 18 results over 11 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