# Lesson Addition and subtraction formulas

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## Addition and subtraction formulas

The addition and subtraction Trigonometry formulas are:

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

### Proof of the addition formula for cosines

In the unit circle consider the point P1 with the central angle
(coordinates (, ), see the Figure 1a).
Consider also the point P2 with the central angle (coordinates
(, ), see the Figure 1a).
Let P3 be the point with the central angle (coordinates
(, ), see the Figure 1b).
We have
, ,   (1)
, ,                   (2)
, .         (3)

Figure 1a. Proof of the addition formula
for cosines

Figure 1b. Proof of the addition formula
for cosines
Since triangles P1OP2 and AOP3 are congruent, the segment [P1,P2] (Figure 1a) has the same length as the segment [A,P3] (Figure 1b),
where A is the point with coordinates (1,0). This gives you the equation
.
Simplify this equation step by step. You get
(after opening the brackets),
(after using , and ),
(after dividing both sides by -2).

Substituting expressions (1), (2) and (3) for , , , , and , you get exactly the addition formula
.

The proof is completed.

### Proof of the subtraction formula for cosines

Now, when the addition formula for cosines is proved, the proof of the subtraction formula can be made in couple of lines. Simply introduce the angle and then apply the addition formula for cosines. Use , :
=
.

The proof is completed.

### Proof of the addition formula for sines

You can easy get the addition formula for sines from the subtraction formula for cosines, which is already proved. Simply use the reduction formulas
,
(see, for example, the lesson The Amazing Unit Circle: Trigonometric Identities of this module).

You have
=
=
.

The proof is completed.

### Proof of the subtraction formula for sines

Similarly, you can easy get the subtraction formula for sines from the addition formula for cosines, which is already proved.
Simply use the same reduction formula as in the previous proof.

=
=
.

The proof is completed.

### Proof of the addition and subtraction formulas for tangents

Now, when the addition and subtraction formulas for cosines and sines are proved, the proof of the addition and subtraction formulas
for tangents is straightforward.

For addition you have

=

=     (after dividing both numerator and denominator by )

= .

The proof is completed.

For subtraction you have

=

=     (after dividing both numerator and denominator by )

= .

The proof is completed.

For examples see the lesson Addition and subtraction formulas - 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.

Addition and subtraction formulas
 , , , , , . The lessons Addition and subtraction formulas (this lesson) 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 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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