Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-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.
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u=v=π/4u=v=\pi/4 shows that the tangent addition formula cannot omit its domain restrictions

Statement refuted

The tangent addition formula cannot be asserted for all real u,vu,v without domain restrictions. The conventions and prerequisite facts used below are recorded in Addition and subtraction formulas for tangent, cotangent, secant, and cosecant on their exact domains, Tangent, cotangent, secant, and cosecant on their exact natural domains, Quarter-turn values and shifts by pi/2 and pi.

Facts & Assumptions

Given: u=v=π/4u=v=\pi/4.

Counterexample

1.1

Both tan⁡u\tan u and tan⁡v\tan v equal 11, so the formal right side has denominator 1−1=01-1=0.

given
2.1

Meanwhile u+v=π/2u+v=\pi/2, where tangent itself is undefined.

given∎

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: 26 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