Trigonometric describes the rules it runs, and its thirty-three identities are written out

[Rafael-SOWNet]

The fourth tranche of the repoint. Trigonometric's comments are notes on soundness — which branch cut, which range, which issue — rather than identities, so all thirty-three were written from the rules' own patterns and replacements.

Two were measured rather than recalled, and one found a wrong answer

arctan(a) + arccotan(a) = pi/2 for a >= 0, and -pi/2 for a < 0

because this library's arccotan has range (-pi/2, pi/2] and not the textbook (0, pi). Measuring that at +1, −1 and 0 rather than recalling it confirmed the identity — and showed Simplify contradicting EvalNumerical on arccotan(-1) two lines up the same output. That is #1113, fixed separately.

arccotan(cotan(a)) = a, for a in arccotan's own range

written that way deliberately, naming no interval. The interval is the thing this codebase has shipped wrong before, and the rule does not need it stated in order to be described.

Forty-three arms become thirty-three rules

	Was	Is
RewriteRules.Trigonometric.Rules.Count	43	33
registry total	407	355
described rules	95	200
sets describing what they run	23	24

Three tests changed, and each was asserting something the repoint improves

• StepAsASentenceTest used Trigonometric as its example of a set still named by rendered patterns. It now finds those sets rather than naming one, since which they are moves with every tranche — the failure cost was the argument for asking instead of naming.
• RewriteRecordingTest.AOneStepRewriteIsOneStep asserted the switch arm's replacement text ("1") for sin²+cos². It now asserts the rule's name and identity, which is what a derivation reports — and its two cases now show the contrast directly: Common named Sumf(var any1, var any1a) when any1 == any1a, Trigonometric named a-squared-sine-and-cosine-of-one-angle-sum-to-one with sin(a)^2 + cos(a)^2 = 1.
• DerivationPathTest.EveryStepNamesWhatDidIt asserted a step's tier equals its set's. It is the weakest of the rewrites that actually fired (A step renders as a sentence #1110), and sin²+cos² is now Sound where Trigonometric is SoundUnderAssumptions — the Pythagorean identity holds for every complex argument, and the set's minimum was understating it. That is the finer grain doing its job, so the test now asserts the documented relation rather than the old one.

What is left

Twenty-four of the thirty sets describe what they run. Three of the six left are the CanonicalOrder family, which still runs its switch. The three others are Common (33 descriptions to port), Power (22) and InequalityEquality (11).

Full suite: 9010 passed, 14 skipped, 0 failed. No public API change.

Part of #746 tier 2 and #825.