# To prove: The subtraction formula for sine function sin ( α − β ) = sin α cos β − cos α sin β by using the addition formula for cosine.

### 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 45E
To determine

## To prove: The subtraction formula for sine function sin(α−β)=sinαcosβ−cosαsinβ by using the addition formula for cosine.

Expert Solution

### Explanation of Solution

It is given that, the addition formula cos(x+y)=cosxcosysinxsiny, cos(π2θ)=sinθ and sin(π2θ)=cosθ.

Substitute x=π2αandy=β in addition formula cos(x+y)=cosxcosysinxsiny.

cos((π2α)+β)=cos(π2α)cosβsin(π2α)sinβcos(π2(αβ))=sinαcosβ+cosαsinβ[cos(π2α)=sinα]sin(αβ)=sinαcosβ+cosαsinβ

That is, sin(αβ)=sinαcosβ+cosαsinβ.

Hence the proof.

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