# To prove : tan ( x + y ) = tan ( x ) + tan ( y ) 1 − tan ( x ) tan ( y )

### 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 20E

(a)

To determine

## To prove : tan(x+y)=tan(x)+tan(y)1−tan(x)tan(y)

Expert Solution

### Explanation of Solution

Given information :

tan(x+y)=tan(x)+tan(y)1tan(x)tan(y)

Formula used :

tan(x)=sin(x)cos(x)cos(x+y)=cos(x)cos(y)sin(x)sin(y)sin(x+y)=sin(x)cos(y)+cos(x)sin(y)

Proof:

Take left hand side:

tan(x+y)=sin(x+y)cos(x+y)(tan(x)=sin(x)cos(x))

Now, apply identities:

cos(x+y)=cos(x)cos(y)sin(x)sin(y)sin(x+y)=sin(x)cos(y)+cos(x)sin(y)tan(x+y)=sin(x)cos(y)+cos(x)sin(y)cos(x)cos(y)sin(x)sin(y)

Dividing the right-hand side by cos(x)cos(y)

tan(x+y)=sin(x)cos(y)cos(x)cos(y)+cos(x)sin(y)cos(x)cos(y)cos(x)cos(y)cos(x)cos(y)sin(x)sin(y)cos(x)cos(y)=sin(x)cos(x)+sin(y)cos(y)1sin(x)sin(y)cos(x)cos(y)=tan(x)+tan(y)1tan(x)tan(y)(tan(x)=sin(x)cos(x))=RHS

Hence, proved

(b)

To determine

Expert Solution

### Explanation of Solution

Given information :

tan(xy)=tan(x)tan(y)1+tan(x)tan(y)

Formula used :

tan(x)=sin(x)cos(x)cos(xy)=cos(x)cos(y)+sin(x)sin(y)sin(xy)=sin(x)cos(y)cos(x)sin(y)

Proof:

Take left hand side:

tan(xy)=sin(xy)cos(xy)(tan(x)=sin(x)cos(x))

Now, apply identities:

cos(xy)=cos(x)cos(y)+sin(x)sin(y)sin(xy)=sin(x)cos(y)cos(x)sin(y)tan(x+y)=sin(x)cos(y)cos(x)sin(y)cos(x)cos(y)+sin(x)sin(y)

Dividing the right-hand side by cos(x)cos(y)

tan(x+y)=sin(x)cos(y)cos(x)cos(y)cos(x)sin(y)cos(x)cos(y)cos(x)cos(y)cos(x)cos(y)+sin(x)sin(y)cos(x)cos(y)=sin(x)cos(x)sin(y)cos(y)1+sin(x)sin(y)cos(x)cos(y)=tan(x)tan(y)1+tan(x)tan(y)(tan(x)=sin(x)cos(x))=RHS

Hence, proved

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