# Lesson Powers of trigonometric functions

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This Lesson (Powers of trigonometric functions) was created by by ikleyn(4)  : View Source, Show

## Powers of trigonometric functions

The formulas for Powers of trigonometric functions are:

,

,

,

.

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

### Proof of the cosines square formula

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
.

to the previous equation. You get
.

Make simple rearrangements in the line above, and you get exactly what we are going to prove.
The proof is completed.

### Proof of the sines square formula

We are going to prove the formula

.

Start from
,

which is kind of the basic formulas.
Substitute
,

the formula which was proved above. You get
.

Make simple rearrangements in the line above, and you get exactly what we are going to prove.
The proof is completed.

### Proof of the cosines cube formula

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 we just have ready the expressions
,
which was proved above.

For the general addition formula for sines gives
.

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

=
=.

Make the last rearrangements in the lines above, and you get exactly what we are going to prove.
The proof is completed.

For examples of applications of these formulas see the lesson Powers of trigonometric functions - 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 (this lesson) and                                                             Powers of Trigonometric functions - Examples

Trigonometric functions of multiply argument
 , , , . The lessons Trigonometric functions of multiply argument 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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