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TrigExpand—Wolfram Documentation

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TrigExpand[expr]

expands out trigonometric functions in expr.

Details
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Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
Possible Issues  
See Also
Tech Notes
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TrigExpand

TrigExpand[expr]

expands out trigonometric functions in expr.

Details

  • TrigExpand operates on both circular and hyperbolic functions.
  • TrigExpand splits up sums and integer multiples that appear in arguments of trigonometric functions, and then expands out products of trigonometric functions into sums of powers, using trigonometric identities when possible.
  • TrigExpand automatically threads over lists, as well as equations, inequalities, and logic functions.

Examples

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Basic Examples  (2)

Expand trigonometric expressions:

Wolfram Language code: TrigExpand[Sin[2x]]
Wolfram Language code: TrigExpand[Sin[x + y]]

Expand hyperbolic trigonometric expressions:

Wolfram Language code: TrigExpand[Tanh[2t]]
Wolfram Language code: TrigExpand[Cosh[x - y]]

Scope  (4)

Expand trigonometric expressions:

Wolfram Language code: TrigExpand[Sin[3x]]
Wolfram Language code: TrigExpand[Cos[x + y + z]]
Wolfram Language code: TrigExpand[Sin[x] ^ 2 + Cos[x] ^ 2]

Expand hyperbolic trigonometric expressions:

Wolfram Language code: TrigExpand[Sinh[4x]]
Wolfram Language code: TrigExpand[Sinh[x - 2y]]

TrigExpand threads over lists:

Wolfram Language code: TrigExpand[{Tan[2x], Sinh[x + y]}]

TrigExpand threads over equations and inequalities:

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

Applications  (1)

Find ArcSin addition law:

Wolfram Language code: Sin[ArcSin[t] + ArcSin[v]]//TrigExpand

Because of the multivaluedness of ArcSin this law does not hold everywhere:

Wolfram Language code: RegionPlot[ArcSin[Sqrt[1 - t^2] v + t Sqrt[1 - v^2]] == ArcSin[t] + ArcSin[v], {t, -1, 1}, {v, -1, 1}]
Wolfram Language code: ArcSin[Sqrt[1 - t^2] v + t Sqrt[1 - v^2]] - (ArcSin[t] + ArcSin[v]) /. {v -> 1, t -> 1}

Properties & Relations  (3)

TrigExpand and TrigReduce are, generically, inverses of each other:

Wolfram Language code: TrigReduce[2Sin[x] ^ 2]
Wolfram Language code: TrigExpand[%]

The expressions are identical modulo trigonometric Pythagorean identities:

Wolfram Language code: % - 2Sin[x] ^ 2

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

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

Use TrigExpand to construct a ChebyshevT polynomial:

Wolfram Language code: TrigExpand[Cos[5x]] /. Sin[x] ^ n_ :> (1 - Cos[x] ^ 2) ^ (n / 2)//Expand
Wolfram Language code: ChebyshevT[5, Cos[x]]

Possible Issues  (1)

Expand trigonometric functions using half-angles:

Wolfram Language code: TrigExpand[Cos[x]]
Wolfram Language code: TrigExpand[Cos[x] + Cos[x / 2]] - Cos[x / 2]
Wolfram Language code: TrigExpand[Cos[x] /. x -> 2t] /. t -> x / 2
Wolfram Research (1996), TrigExpand, Wolfram Language function, https://reference.wolfram.com/language/ref/TrigExpand.html (updated 2007).

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_trigexpand, author="Wolfram Research", title="{TrigExpand}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/TrigExpand.html}", note=[Accessed: 14-November-2026]}

BibLaTeX

@online{reference.wolfram_2026_trigexpand, organization={Wolfram Research}, title={TrigExpand}, year={2007}, url={https://reference.wolfram.com/language/ref/TrigExpand.html}, note=[Accessed: 14-November-2026]}