# Lesson Product of trigonometric functions

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## Product of trigonometric functions

The formulas for the product of trigonometric functions are:

,

,

.

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

### Proof of the sines product formula

We are going to prove the formula

.

The proof is very simple and straightforward.

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

,
.

Subtract the first formula from the second one side by side. You get

.

Now, divide both sides of the last equality by 2.
You got what we were going to prove. The proof is completed.

### Proof of the cosines product formula

We are going to prove the formula

.

Use the same addition and subtraction formulas for cosines of the lesson Addition and subtraction formulas in this module:

,
.

Now, add the first formula to the second one side by side. You get

.

Divide both sides of the last equality by 2.
You got what we were going to prove. The proof is completed.

### Proof of the sines-cosines product formula

We are going to prove the formula

,

Now, use the addition and subtraction formulas for sines of the lesson Addition and subtraction formulas of this module:

,
.

Add the first formula to the second one side by side. You get

.

Divide both sides of the last equality by 2.
You got what we were going to prove. The proof is completed.

For examples of applications of these formulas see the lesson Product 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 (this lesson) 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 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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