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TrigFactor[expr]

factors trigonometric functions in expr.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
Possible Issues  
See Also
Tech Notes
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TrigFactor

TrigFactor[expr]

factors trigonometric functions in expr.

Details

  • TrigFactor operates on both circular and hyperbolic functions.
  • TrigFactor splits up sums and integer multiples that appear in arguments of trigonometric functions, and then factors resulting polynomials in trigonometric functions, using trigonometric identities when possible.
  • TrigFactor automatically threads over lists, as well as equations, inequalities, and logic functions.

Examples

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Basic Examples  (3)

Factor a trigonometric expression into a product of terms:

Wolfram Language code: TrigFactor[Sin[x] ^ 2 + Tan[x] ^ 2]

Factor a hyperbolic trigonometric expression:

Wolfram Language code: TrigFactor[Cosh[x] ^ 2 - Cosh[x] ^ 4]

Factor trigonometric expressions of several variables:

Wolfram Language code: TrigFactor[Sin[x + y] ^ 2 + Tan[x + y]]

Scope  (6)

Factor a univariate trigonometric expression:

Wolfram Language code: TrigFactor[Sin[x] ^ 2 - Cos[x] ^ 2]

Factor a multivariate trigonometric expression:

Wolfram Language code: TrigFactor[Cos[x + y] + Sin[x]Sin[y]]

Factor a hyperbolic trigonometric expression:

Wolfram Language code: TrigFactor[Cosh[x] ^ 2 - Cosh[x] ^ 4]

Factor a rational combination of trigonometric functions:

Wolfram Language code: TrigFactor[Cosh[x + y] / (Sin[x] ^ 2 - Cos[x] ^ 2) + 1 / Sinh[x]]

TrigFactor threads over lists:

Wolfram Language code: TrigFactor[{Sin[x] + Cos[x], Sinh[x] ^ 2 + Tanh[x] ^ 2}]

TrigFactor threads over equations and inequalities:

Wolfram Language code: TrigFactor[1 < Cos[x] + Cos[y] < 2]

Applications  (2)

Solve a trigonometric polynomial equation:

Wolfram Language code: TrigFactor[Cos[x]^3 + 3 Cos[x]^2 Sin[x] + 3 Cos[x] Sin[x]^2 + Sin[x]^3 == 2]

Reduce it to elementary trigonometric equations:

Wolfram Language code: Reduce[%, Sin[Pi / 4 + x]]

Detect common roots of trigonometric polynomials:

Wolfram Language code: p = Sin[x] ^ 2 + Tan[x] ^ 2; q = -2Cos[x] ^ 2 + Sin[2x] ^ 2 / 4;
Wolfram Language code: TrigFactor[p]
Wolfram Language code: TrigFactor[q]

Common roots of and are the roots of :

Wolfram Language code: Solve[3 + Cos[2 x] == 0, x]

Using the original polynomials yields a more complicated result:

Wolfram Language code: Solve[p == 0 && q == 0, x]

To show that the root sets are equal, it suffices to check within one period:

Wolfram Language code: FunctionPeriod[{p, q}, x]
Wolfram Language code: Solve[3 + Cos[2 x] == 0 && 0 ≤ Re[x] < Pi, x]
Wolfram Language code: Solve[p == 0 && q == 0 && 0 ≤ Re[x] < Pi, x]
Wolfram Language code: FullSimplify[TrigToExp[(x /. %%) - (x /. %)]]

Properties & Relations  (2)

Compare TrigExpand, TrigReduce, and TrigFactor on the same expression:

Wolfram Language code: TrigFactor[Sin[x] ^ 2 + Tan[x] ^ 2]
Wolfram Language code: TrigExpand[Sin[x] ^ 2 + Tan[x] ^ 2]
Wolfram Language code: TrigReduce[Sin[x] ^ 2 + Tan[x] ^ 2]

TrigFactor threads element‐wise over lists, inequalities, equations, and logical operations:

Wolfram Language code: TrigFactor[Sin[x] + Cos[x] == 0]

Possible Issues  (1)

A trigonometric expression can be factored in many different ways:

Wolfram Language code: TrigFactor[Sin[2x] + 2Cos[x]]
Wolfram Language code: ExpToTrig[Factor[TrigToExp[Sin[2x] + 2Cos[x]] /. x -> I Log[t]] /. t -> E ^ (-I x)]
Wolfram Language code: Simplify[%% - %]
Wolfram Language code: 2Sin[x] + 3Cos[x] /. (a_.)Sin[x_] + (b_.)Cos[x_] /; Positive[a] && Positive[b] :> Sqrt[a ^ 2 + b ^ 2]Cos[x - ArcCos[b / Sqrt[a ^ 2 + b ^ 2]]]
Wolfram Language code: Simplify[% - (2Sin[x] + 3Cos[x])]
Wolfram Research (1996), TrigFactor, Wolfram Language function, https://reference.wolfram.com/language/ref/TrigFactor.html (updated 2007).

Text

Wolfram Research (1996), TrigFactor, Wolfram Language function, https://reference.wolfram.com/language/ref/TrigFactor.html (updated 2007).

CMS

Wolfram Language. 1996. "TrigFactor." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/TrigFactor.html.

APA

Wolfram Language. (1996). TrigFactor. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TrigFactor.html

BibTeX

@misc{reference.wolfram_2026_trigfactor, author="Wolfram Research", title="{TrigFactor}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/TrigFactor.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_trigfactor, organization={Wolfram Research}, title={TrigFactor}, year={2007}, url={https://reference.wolfram.com/language/ref/TrigFactor.html}, note=[Accessed: 13-June-2026]}