# Lesson Trigonometric functions of multiply argument

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## Trigonometric functions of multiply argument

The formulas for the Trigonometric functions of multiply argument are:

,

,

,

.

In this lesson you can learn how to prove these formulas.

### Proof of the cosine formula for the double argument

We are going to prove the formula

.

The proof is very simple and straightforward.

It is based on the addition formula for cosines of the lesson Addition and subtraction formulas in this module:

.

Simply take in this formula. You get

.

Now, substitute here
.

You get
.

This is exactly what we are going to prove.
The proof is completed.

### Proof of the sines formula for the double argument

We are going to prove the formula

.

It is based on the addition formula for sines of the lesson Addition and subtraction formulas in this module:

.

Take in this formula. You get
,

exactly what we are going to prove.
The proof is completed.

### Proof of the cosine formula for the triple argument

We are going to prove the formula

Let us apply the addition formula for cosines of the lesson Addition and subtraction formulas of this module
in the form

. (*)

For and we just have proved the formulas
,

.

Now, substitute these expressions for and to the formula (*) above. You get

=
=.

This is exactly what we are going to prove.
The proof is completed.

For examples of the applications of these formulas see the lesson Trigonometric functions of multiply argument - Examples in this module.

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For your convenience, below is the list of my lessons on Trigonometry in this site in the logical order.
They all are under the current topic Trigonometry in the section Algebra II.

 , , , , , . The lessons Addition and subtraction formulas and                      Addition and subtraction formulas - Examples

Addition and subtraction of trigonometric functions
 , , , , , . The lessons Addition and subtraction of trigonometric functions and                      Addition and subtraction of trigonometric functions - Examples

Product of trigonometric functions
 , , . The lessons Product of trigonometric functions and                                                    Product of trigonometric functions - Examples

Powers of trigonometric functions
 , , , . The lessons Powers of Trigonometric functions and                                                             Powers of Trigonometric functions - Examples

Trigonometric functions of multiply argument
 , , , . The lessons Trigonometric functions of multiply argument (this lesson) and                                                                 Trigonometric functions of multiply argument - Examples

Trigonometric functions of half argument
 , , , , , . The lessons Trigonometric functions of half argument and                   Trigonometric functions of half argument - Examples

Miscellaneous Trigonometry problems

The lesson Miscellaneous Trigonometry problems

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