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 26E
To determine

To prove:

  cos3θ=4cos3θ3cosθ

Expert Solution

Explanation of Solution

Given information:

Eq: cos3θ=4cos3θ3cosθ

Formula Used:

Subtraction formula:

  cos(xy)=cosxcosy+sinxsiny

The double angle formulas:

  cos2x=2cos2x1sin2x=2sinxcosx

Proof:

The equation is given as:

  cos3θ=4cos3θ3cosθ

The left hand side of the eq.:

   cos3θ=cos( 4θθ )

   cos3θ=cos4θcosθ+sin4θsinθ

   cos3θ=( 2 cos 2 2θ1 )cosθ+( 2sin2θcos2θ )sinθ[ 2 cos 2 θ1=cos2θ sin2θ=2sinθcosθ ]

   cos3θ=( 2 ( 2 cos 2 θ1 ) 2 1 )cosθ+( 2sin2θcos2θ )sinθ

   cos3θ=( 2( 4 cos 4 θ4 cos 2 θ+1 )1 )cosθ+( 2sin2θcos2θ )sinθ[ ( ab ) 2 = a 2 2ab+ b 2 ]

   cos3θ=( 8 cos 4 θ8 cos 2 θ+21 )cosθ+( 2( 2sinθcosθ )( 2 cos 2 θ1 ) )sinθ

   cos3θ=( 8 cos 5 θ8 cos 3 θ+cosθ )+( 4 sin 2 θcosθ )( 2 cos 2 θ1 )

   cos3θ=( 8 cos 5 θ8 cos 3 θ+cosθ )+( 4( 1 cos 2 θ )cosθ )( 2 cos 2 θ1 )[ sin 2 θ+ cos 2 θ=1 ]

   cos3θ=( 8 cos 5 θ8 cos 3 θ+cosθ )+( 4cosθ )( 1 cos 2 θ )( 2 cos 2 θ1 )

   cos3θ=( 8 cos 5 θ8 cos 3 θ+cosθ )+( 4cosθ )( 2 cos 2 θ12 cos 4 θ+ cos 2 θ )

   cos3θ=( 8 cos 5 θ8 cos 3 θ+cosθ )+( 4cosθ )( 3 cos 2 θ12 cos 4 θ )

   cos3θ=8 cos 5 θ8 cos 3 θ+cosθ+12 cos 3 θ4cosθ8 cos 5 θ

   cos3θ=4 cos 3 θ3cosθ

Since, left hand side (cos3θ) equals to right hand side (4cos3θ3cosθ) for all values θ . So, cos3θ=4cos3θ3cosθ is an identity.

Hence, proved.

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