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.

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The tangent half-angle identities and rational parametrization of the unit circle away from $(- 1, 0)$

Statement

If $t = \tan (θ / 2)$ is defined, then $\cos θ = \frac{1 - t^{2}}{1 + t^{2}}, \sin θ = \frac{2 t}{1 + t^{2}} .$ Conversely every point $(x, y)$ on $x^{2} + y^{2} = 1$ with $x \neq - 1$ has the unique parameter $t = y / (1 + x)$ and equals $((1 - t^{2}) / (1 + t^{2}), 2 t / (1 + t^{2}))$ . The conventions and prerequisite facts used below are recorded in Double-angle and quadratic power-reduction identities, Tangent, cotangent, secant, and cosecant on their exact natural domains, Parity and the Pythagorean identity for sine and cosine.

Facts & Assumptions

Given: The indicated real parameter or unit-circle point.

Proof

technique · direct

1.1

Divide the double-angle identities by $\cos^{2} (θ / 2)$ to obtain the rational formulas; $1 + t^{2} > 0$ .

algebra

1.2

For a point with $x \neq - 1$ , put $t = y / (1 + x)$ and use $x^{2} + y^{2} = 1$ to simplify both rational expressions to $x$ and $y$ .

algebra

2.1

The same formula $t = y / (1 + x)$ proves uniqueness.

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: 32 results over 19 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)

The tangent half-angle identities and rational parametrization of the unit circle away from (−1,0)(−1,0)