Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16
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 addition formulas for complex trigonometric and hyperbolic functions

Statement

For all z,w∈C, sin⁡(z+w)=sin⁡zcos⁡w+cos⁡zsin⁡w, cos⁡(z+w)=cos⁡zcos⁡w−sin⁡zsin⁡w, sinh⁡(z+w)=sinh⁡zcosh⁡w+cosh⁡zsinh⁡w, cosh⁡(z+w)=cosh⁡zcosh⁡w+sinh⁡zsinh⁡w.

Facts & Assumptions

Given: Complex numbers z,w.

[L1]

The four functions are defined by their symmetric and antisymmetric exponential combinations (Complex sine, cosine, hyperbolic sine, and hyperbolic cosine from the complex exponential).

[L2]

For all complex u,v, exp⁡(u+v)=exp⁡uexp⁡v (exp⁡(z+w)=exp⁡z exp⁡w, and the complex exponential extends the real exponential).

Proof

technique · direct
1.1

Substitute z+w into the four formulas in [L1] and factor every exponential using [L2].

L1L2
2.1

Expanding the right-hand sides in the Statement with [L1] gives the same symmetric or antisymmetric combinations, so the four identities follow.

step 1.1L1algebra∎

Depends on

Used by

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Sources