DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 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.
Tangent, cotangent, secant, and cosecant on their exact natural domains
Definition
Define where ranges over the integers. The exclusions are exactly the zero sets of The zero sets of sine and cosine and the least positive common period 2 pi.
Depends on
Used by
- u=v=π/4 shows that the tangent addition formula cannot omit its domain restrictions Counterexample
- Addition and half-angle identities compute the sine, cosine, and tangent of π/12 Example
- Machin's formula π/4=4arctan(1/5)-arctan(1/239) Example
- Tangent is a continuous strictly increasing bijection from (-π/2,π/2) onto ℝ Lemma
- Addition and subtraction formulas for tangent, cotangent, secant, and cosecant on their exact domains Theorem
- Cofunction, supplementary, quarter-turn, and reflection identities for the six trigonometric functions Theorem
- Derivatives and fundamental periods of tangent, cotangent, secant, and cosecant Theorem
- Double-angle and quadratic power-reduction identities Theorem
- Half-angle identities with the sign determined by the quadrant Theorem
- Principal arctangent: derivative, integral, power series, and the Gregory–Leibniz series Theorem
- Pythagorean and parity identities for all six trigonometric functions on their natural domains Theorem
- The tangent half-angle identities and rational parametrization of the unit circle away from (-1,0) Theorem
- Triple-angle identities for sine, cosine, and tangent Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 35 results over 10 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
- NIST Digital Library of Mathematical Functions, Chapter 4 (standard reference, not scraped)
- C. Schmeiser, Introduction to Analysis (standard reference, not scraped)