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TrigFactor

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

TrigFactor[expr]
    factors trigonometric functions in expr.
    TrigFactor operates on both circular and hyperbolic functions.
    TrigFactor factors polynomials in trigonometric functions and collapses
    Pythagorean, angle-addition, and double-angle identities where possible,
    broadly acting as the inverse of TrigExpand.
    TrigFactor automatically threads over lists, as well as equations,
    inequalities, and logic functions.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= TrigFactor[Sin[x]^2 + Cos[x]^2]
Out[1]= 1

In[2]:= TrigFactor[Cosh[x]^2 - Sinh[x]^2]
Out[2]= 1

In[3]:= TrigFactor[2 Sin[x] Cos[x]]
Out[3]= Sin[2 x]

In[4]:= TrigFactor[Cos[x]^2 - Sin[x]^2]
Out[4]= Cos[2 x]

In[5]:= TrigFactor[Sin[a] Cos[b] + Cos[a] Sin[b]]
Out[5]= Sin[a + b]

In[6]:= TrigFactor[Cos[a] Cos[b] + Sin[a] Sin[b]]
Out[6]= Cos[a - b]

In[7]:= TrigFactor[Sin[x]^2 + Tan[x]^2]
Out[7]= (1 + Cos[x]^2) Tan[x]^2

In[8]:= TrigFactor[Cosh[x]^2 - Cosh[x]^4]
Out[8]= -Cosh[x]^2 Sinh[x]^2

Implementation notes

Algorithm. builtin_trigfactor_impl factors trig expressions — broadly the inverse of TrigExpand for the structural identities both support. It tries two parallel paths and keeps the shorter result (by trigfactor_leaf_count):

  • Path A: trigfactor_run_pipeline on the argument as-is. If this changes the expression at all it is trusted and Path B is skipped (Path B can be expensive on angle-sum arguments).
  • Path B: only attempted when Path A was a no-op and the input has compound trig structure (has_compound_trig_structure — a Power[trig, k≥2] or a Times of two or more trig atoms). First TrigExpand the argument, then run the pipeline; kept only if strictly shorter than Path A. This catches cancellations that surface only after angle-sum expansion (e.g. Cos[x+y] + Sin[x] Sin[y] -> Cos[x] Cos[y]).

trigfactor_run_pipeline (trig canonicalizer suppressed throughout): 1. ReplaceRepeated with trig_factor_to_sincos — rewrite reciprocal heads as Sin/Cos ratios so Factor sees full polynomial structure. 2. Together over a common denominator. 3. Factor — Mathilda Factor treats trig atoms as polynomial variables. Skipped (left as the Together output) when the polynomial would stall: more than TRIG_FACTOR_ATOM_THRESHOLD distinct squared atoms, max trig-atom power above TRIG_FACTOR_DEGREE_THRESHOLD, or more than TRIG_FACTOR_TOTAL_ATOM_THRESHOLD distinct atoms. 4. ReplaceRepeated with trig_factor_identities — Pythagorean collapses (both signs, circular and hyperbolic), reverse angle-addition, reverse double-angle, factored-form (Cos-Sin)(Cos+Sin) -> Cos[2x] and (Cosh±Sinh) collapses, and the Weierstrass-style linear-combination factoring a Sin[x] + b Cos[x] -> Sqrt[a^2+b^2] Sin[x + ArcTan[a, b]] (gated by NumberQ and Im == 0 on both coefficients so complex coefficients cannot collapse Sqrt[a^2+b^2] to zero). 5. ReplaceRepeated with trig_factor_from_sincos to restore Tan/Sec/...

Data structures. Static rule lists from trigsimp_init; tail patterns (r___) on Plus/Times let each identity fire inside a larger sum/product, with ATTR_ORDERLESS on Plus/Times driving the permutation search. Memoized through the active FactorMemo via the builtin_trigfactor wrapper.

  • Listable, Protected.
  • Operates on both circular (Sin, Cos, Tan, Cot, Sec, Csc) and

Attributes: Listable, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

TrigFactor collapses an expanded angle-addition expression back to a single trigonometric function:

In[1]:= TrigFactor[Sin[a] Cos[b] + Cos[a] Sin[b]]
Out[1]= Sin[a + b]

In[1]:= TrigFactor[Cos[a] Cos[b] - Sin[a] Sin[b]]
Out[1]= Cos[a + b]

A difference of squares of sine and cosine folds into a double angle:

In[1]:= TrigFactor[Sin[x]^2 - Cos[x]^2]
Out[1]= -Cos[2 x]

A full trigonometric perfect square is factored and phase-shifted into a single squared sine:

In[1]:= TrigFactor[Sin[x]^2 + 2 Sin[x] Cos[x] + Cos[x]^2]
Out[1]= 2 Sin[1/4 Pi + x]^2

It handles hyperbolic functions too, recognising the cosh double-angle identity:

In[1]:= TrigFactor[Sinh[x]^2 + Cosh[x]^2]
Out[1]= Cosh[2 x]