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.

The subtraction formulas for sine and cosine

For all real $u, v$ , $\sin (u - v) = \sin u \cos v - \cos u \sin v, \cos (u - v) = \cos u \cos v + \sin u \sin v .$ 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.

Given: Reals $u, v$ .

technique · direct

1.1given

Apply the addition formulas to $u + (- v)$ .

2.1algebra∎

Replace $\sin (- v)$ and $\cos (- v)$ by their parity values and simplify.

Depends on