BuyFind

Single Variable Calculus: Concepts...

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

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)1tan(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

To prove : tan(xy)=tan(x)tan(y)1+tan(x)tan(y)

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

Have a homework question?

Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!