# 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:

Reduce hyperbolic trigonometric expressions:

## Scope(4)

Trigonometric expressions:

Hyperbolic trigonometric expressions:

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: