# Lesson Addition and subtraction of trigonometric functions

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## Addition and subtraction of trigonometric functions

The addition and subtraction trigonometric functions formulas are:

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

### Proof of the sines addition formula

We are going to prove the formula

.

The proof is very simple and straightforward.

Introduce two new angles by formulas
, .

Note that
,

.

Now, you can express as using the addition formula for sines:
(this addition formula was proved in the lesson Addition and subtraction formulas in this module)

and express as using the subtraction formula for sines:
(this subtraction formula was proved in the lesson Addition and subtraction formulas in this module).

Add left sides and right sides of these two formulas. You get
,

what is exactly what we were going to prove.

### Proof of the sines subtraction formula

We are going to prove the formula

.

The proof is very similar to the previous one.

Use the same angles and as before. Express as using the same addition formula for sines:

and express as using the same subtraction formula for sines:
.

Now, subtract the second formula from the first one side by side.You get
,

which is exactly what we were going to prove.

Proofs of the addition and subtraction formulas for cosines are very similar to that of sines.

### Proof of the tangents addition formula

We are going to prove the formula

Start from the left side and transform it step by step:
(after converting the fractions to the common denominator)

=                                             (after using the formula of sines for the sum of angles)

This is exactly what we were going to prove.

### Proof of the tangents subtraction formula

The proof is similar to the previous one.

For examples of applications of these formulas see the lesson Addition and subtraction 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 (this lesson) 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 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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