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 24E
To determine

To prove:

  (sinx+cosx)2=1+sin2x

Expert Solution

Explanation of Solution

Given information:

Eq: (sinx+cosx)2=1+sin2x

Formula Used:

The double angle formula:

  sin2x=2sinxcosx

Proof:

The equation is given as:

  (sinx+cosx)2=1+sin2x

The left hand side of the eq.:

  (sinx+cosx)2=(sinx+cosx)(sinx+cosx)(sinx+cosx)2=(sinx)(sinx)+(sinx)(cosx)+(cosx)(sinx)+(cosx)(cosx)(sinx+cosx)2=sin2x+cos2x+2sinxcosx(sinx+cosx)2=1+2sinxcosx[sin2x+cos2x=1](sinx+cosx)2=1+sin2x[sin2x=2sinxcosx]

Since, left hand side (sinx+cosx)2 equals to right hand side (1+sin2x) for all values x . So, (sinx+cosx)2=1+sin2x is an identity.

Hence, proved.

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