Reduce the double angle over the single one (#557)

[Rafael-SOWNet]

Closes the last part of #557.

"sin(2*x) * cosec(x)".Simplify()   // master: csc(x) * sin(2 * x)
"(sin(2*x)*cosec(x))^2/4 - cos(2*x) - sin(x)^2".Simplify()
// master: (sin(2 * x) * csc(x)) ^ 2 / 4 - cos(x) ^ 2

That second one is the expression from the issue, and it is zero.

Why it stopped one step short

sin(2u) csc(u) is 2 cos(u). Opening sin(2u) up — which the multiple-angle expansion from the earlier part of this issue does — leaves 2 sin(u) cos(u) csc(u), whose sine and cosecant are no longer adjacent in the product. The rules that cancel those match a pair of factors, so they never meet.

Hence a rule of its own, matching before the double angle is opened and the pieces are scattered.

The condition is carried

2cos(u) is a number where sin(u) is zero and sin(2u) csc(u) is not, so an unconditional answer would speak for a point the expression does not reach:

sin(2x) * cosec(x)  →  2 * cos(x) provided not sin(x) = 0
sin(x)  * cosec(x)  →  1          provided not sin(x) = 0     (already the case)

Same condition the neighbouring cancellation already comes back with.

Verification

17 tests, 7 of which fail without the change — including that the rewrite is the same number wherever both sides are defined, checked at six points. Nothing outside the identity is claimed: a different argument (sin(2x) csc(y)) and a multiplier it does not cover (sin(3x) csc(x)) are left alone and pinned as such.

4026 unit tests and 127 F# tests on both target frameworks, none failing. 117-problem self-verifying corpus at 102/117 — simp:trig goes from 67% to 100% — with nothing wrong, in error or timing out. 1320 property checks over 151 expressions, none failing.

[codecov-commenter]

Codecov Report

❌ Patch coverage is 86.66667% with 2 lines in your changes missing coverage. Please review.
✅ Project coverage is 81.60%. Comparing base (90c00a8) to head (06ed34e).
⚠️ Report is 74 commits behind head on master.

Files with missing lines	Patch %	Lines
...s/Simplification/Patterns/Patterns.Trigonometry.cs	86.66%	2 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #693      +/-   ##
==========================================
+ Coverage   80.99%   81.60%   +0.60%     
==========================================
  Files         155      157       +2     
  Lines       13687    13362     -325     
  Branches     1957     2200     +243     
==========================================
- Hits        11086    10904     -182     
+ Misses       1990     1851     -139     
+ Partials      611      607       -4     

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[Happypig375]

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