# To prove: The subtraction formula cos ( α − β ) = cos α cos β + sin α sin β by the use of given figure.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter C, Problem 43E
To determine

## To prove: The subtraction formula cos(α−β)=cosαcosβ+sinαsinβ by the use of given figure.

Expert Solution

### Explanation of Solution

From the given figure, it is observed that A(cosα,sinα)andB(cosβ,sinβ) are two distinct point and c is the distance from A to B.

Obtain the value of c2 by law of cosine.

c2=12+122(1)(1)cos(αβ)=2(1cos(αβ))

That is, c2=2(1cos(αβ)) (1)

The distance between any two point (x1,x2)and(y1,y2) is (x1y1)2+(x2y2)2.

Thus, the distance c from (cosα,sinα) to (cosβ,sinβ) is c=(cosαcosβ)2+(sinαsinβ)2.

Further simplified as,

c2=(cosαcosβ)2+(sinαsinβ)2=cos2α+cos2β2cosαcosβ+sin2α+sin2β2sinαsinβ=(cos2α+sin2α)+(cos2β+sin2β)2cosαcosβ2sinαsinβ=22cosαcosβ2sinαsinβ

That is, c2=22cosαcosβ2sinαsinβ (2)

From equation (1) and (2) it is observed that,

2(1cos(αβ))=22cosαcosβ2sinαsinβ

Further simplified as,

2(1cos(αβ))=22cosαcosβ2sinαsinβ2(1cos(αβ))=2(1cosαcosβsinαsinβ)1cos(αβ)=1cosαcosβsinαsinβcos(αβ)=cosαcosβsinαsinβ

That is, cos(αβ)=cosαcosβ+sinαsinβ.

Hence the proof.

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