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.


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Basic Examples  (2)

Reduce trigonometric expressions:

Reduce hyperbolic trigonometric expressions:

Scope  (4)

Trigonometric expressions:

Hyperbolic trigonometric expressions:

TrigReduce threads over lists:

TrigReduce threads over equations, inequalities and logical operations:

Options  (1)

Modulus  (1)

Manipulation with polynomials is performed using modular arithmetic:

Compare with the reduction over rationals:

Applications  (1)

Find the period of a trigonometric polynomial:

FunctionPeriod gives a multiple of the minimal period:

Reducing the expression helps to find the minimal period:

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

Properties & Relations  (3)

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

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

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

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

Possible Issues  (3)

The value of the option Modulus must be an integer:

TrigReduce requires explicit trigonometric functions:

Use ExpToTrig to convert exponential to trigonometric functions:

Reducing constants might not always give the desired effect:

Neat Examples  (1)

Wolfram Research (1996), TrigReduce, Wolfram Language function, (updated 2007).


Wolfram Research (1996), TrigReduce, Wolfram Language function, (updated 2007).


Wolfram Language. 1996. "TrigReduce." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007.


Wolfram Language. (1996). TrigReduce. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_trigreduce, author="Wolfram Research", title="{TrigReduce}", year="2007", howpublished="\url{}", note=[Accessed: 25-May-2024 ]}


@online{reference.wolfram_2024_trigreduce, organization={Wolfram Research}, title={TrigReduce}, year={2007}, url={}, note=[Accessed: 25-May-2024 ]}