# To prove: cos 3 θ = 4 cos 3 θ − 3 cos θ

### 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 26E
To determine

## To prove:   cos3θ=4cos3θ−3cosθ

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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