Sources and method

These are standard identities of school mathematics, not original results. What this site adds is the structure around them: each formula in its own address, a proof and a derivation, and a machine check of the algebra before anything is published.

Named sources

• OpenStax Precalculus (free textbook) — Chapter 7, Trigonometric Identities and Equations: the reciprocal/quotient/Pythagorean group (7.1), sum and difference identities (7.2), double-angle, half-angle and reduction identities (7.3), law of sines/cosines context (7.4).
• Paul's Online Math Notes — Trig Cheat Sheet — Independent cross-check that the published list is the standard set and that no family is missing at this level.
• Wikipedia — List of trigonometric identities — Used only as a completeness check of the catalogue, never as a source of wording or of values.
• This site's own numerical sampling — Every identity in data/out/identities.json was evaluated at up to 800 random angles with a relative tolerance of 1e-9 before it could be published.

What was not copied

No text, table layout or worked example on this site is taken from another publisher. Mathematical statements are not copyrightable, but sentences are — the wording here was written for these pages, and the proofs are line-by-line reconstructions rather than transcriptions.

What the machine checks

• Every identity holds at up to 800 sampled angles (1e-9 relative tolerance).
• A deliberately broken right-hand side must FAIL that check — otherwise the test itself is blind.
• Conditional identities must fail OUTSIDE their stated interval, which is how the half-angle sign condition was found.
• Triangle relations must fail with six independent random numbers, and pass only with real triangles (A+B+C = 180°).
• Slugs, category references, related links and example links must all resolve.
• Any failure stops the build: no partial catalogue is ever published.

Published data: identities.json, examples.json, index.json (version 2026.09.26-1, checked 2026-09-26).