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TrigReduce[expr]

rewrites products and powers of trigonometric functions in expr in terms of trigonometric functions with combined arguments.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
Modulus  
Applications  
Properties & Relations  
Possible Issues  
Neat Examples  
See Also
Tech Notes
Related Guides
History
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TrigReduce

TrigReduce[expr]

rewrites products and powers of trigonometric functions in expr in terms of trigonometric functions with combined arguments.

Details and Options

  • TrigReduce operates on both circular and hyperbolic functions.
  • Given a trigonometric polynomial, TrigReduce typically yields a linear expression involving trigonometric functions with more complicated arguments.
  • TrigReduce automatically threads over lists, as well as equations, inequalities and logic functions.

Examples

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Basic Examples  (2)

Reduce trigonometric expressions:

Wolfram Language code: TrigReduce[2Cos[x] ^ 2]
Wolfram Language code: TrigReduce[2Sin[x]Cos[y]]

Reduce hyperbolic trigonometric expressions:

Wolfram Language code: TrigReduce[2Cosh[x] ^ 2]
Wolfram Language code: TrigReduce[2Sinh[x]Cosh[y]]

Scope  (4)

Trigonometric expressions:

Wolfram Language code: TrigReduce[2Sin[x + y]Cos[x - y]]
Wolfram Language code: TrigReduce[Tan[x] + Tan[y]]

Hyperbolic trigonometric expressions:

Wolfram Language code: TrigReduce[2Cosh[x]Cosh[y]]
Wolfram Language code: TrigReduce[Coth[x] + Coth[y]]

TrigReduce threads over lists:

Wolfram Language code: TrigReduce[{Tan[x] + Cot[y], Tanh[x] - Coth[y]}]

TrigReduce threads over equations, inequalities and logical operations:

Wolfram Language code: TrigReduce[4 Sin[x] ^ 4 == 1 && 2Cos[x] ^ 2 ≥ 1]

Options  (1)

Modulus  (1)

Manipulation with polynomials is performed using modular arithmetic:

Wolfram Language code: Cos[x] ^ 50
Wolfram Language code: TrigReduce[%, Modulus -> 7]

Compare with the reduction over rationals:

Wolfram Language code: TrigReduce[%%]
Wolfram Language code: Collect[%, _Cos, Mod[Numerator[#]PowerMod[Denominator[#], -1, 7], 7]&]

Applications  (1)

Find the period of a trigonometric polynomial:

Wolfram Language code: f = Cos[2 x] Cos[7 x] Sin[x] + Cos[x] Cos[7 x] Sin[2 x] + Cos[3 x] Sin[7 x] + Cos[8 x] Sin[7 x] + Cos[7 x] Sin[8 x];

FunctionPeriod gives a multiple of the minimal period:

Wolfram Language code: FunctionPeriod[f, x]
Wolfram Language code: Plot[f, {x, 0, %}]

Reducing the expression helps to find the minimal period:

Wolfram Language code: TrigReduce[f]
Wolfram Language code: p = FunctionPeriod[%, x]

Periodicity can also be observed from the plots of the original function and the shifted function:

Wolfram Language code: Animate[Plot[{Evaluate[(f /. x -> x + per)], f }, {x, 0, Pi}], {{per, p}, 0, Pi, Appearance -> "Labeled"}, SaveDefinitions -> True]

Properties & Relations  (3)

ChebyshevT[n,Cos[x]] reduces to Cos[n x]:

Wolfram Language code: ChebyshevT[10, Cos[x]]
Wolfram Language code: TrigReduce[%]

ChebyshevU[n,Cos[x]] is related to Sin[n x]:

Wolfram Language code: TrigReduce[Sin[x]ChebyshevU[10, Cos[x]]]

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

Wolfram Language code: TrigExpand[Sin[5x]]
Wolfram Language code: TrigReduce[%]

TrigReduce threads over lists, inequalities, equations and logical operations:

Wolfram Language code: TrigReduce[4 Sin[x] ^ 4 == 1 && 2Cos[x] ^ 2 ≥ 1]

Possible Issues  (3)

The value of the option Modulus must be an integer:

Wolfram Language code: TrigReduce[Sinh[t] ^ 2, Modulus -> Pi]

TrigReduce requires explicit trigonometric functions:

Wolfram Language code: TrigReduce[(Exp[I x] + Exp[-I x]) ^ 3]

Use ExpToTrig to convert exponential to trigonometric functions:

Wolfram Language code: ExpToTrig[%]//TrigReduce

Reducing constants might not always give the desired effect:

Wolfram Language code: TrigReduce[Cosh[1] ^ 3]
Wolfram Language code: TrigReduce[Cosh[x] ^ 3] /. x -> 1

Neat Examples  (1)

Reduce a product of :

Wolfram Language code: 4 Product[Cos[2 ^ k x], {k, 0, 2}]
Wolfram Language code: TrigReduce[%]
Wolfram Research (1996), TrigReduce, Wolfram Language function, https://reference.wolfram.com/language/ref/TrigReduce.html (updated 2007).

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_trigreduce, author="Wolfram Research", title="{TrigReduce}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/TrigReduce.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_trigreduce, organization={Wolfram Research}, title={TrigReduce}, year={2007}, url={https://reference.wolfram.com/language/ref/TrigReduce.html}, note=[Accessed: 13-June-2026]}