Prove that cos(2∏/7), cos(4∏/7), cos(6∏/7) are the roots of this equation.

[Michael_Light]

Homework Statement

When cos(4θ)=cos(3θ). prove that θ=0, 2∏/7, 4∏/7, 6∏/7

Hence prove that cos(2∏/7), cos(4∏/7), cos(6∏/7) are the roots of 8x³+4x²-4x-1

Homework Equations

The Attempt at a Solution

I can do the first part, but i have some difficulty in solving the second part. For the second part, I start by letting x= cosθ and try to solve the equation, however, i notice that i couldn't simplify the equation... so it is correct to let x=cosθ? How is the second part related to the first part? Thanks in advance.

 

[Dick]

Homework Statement

When cos(4θ)=cos(3θ). prove that θ=0, 2∏/7, 4∏/7, 6∏/7

Hence prove that cos(2∏/7), cos(4∏/7), cos(6∏/7) are the roots of 8x³+4x²-4x-1

Homework Equations

The Attempt at a Solution

I can do the first part, but i have some difficulty in solving the second part. For the second part, I start by letting x= cosθ and try to solve the equation, however, i notice that i couldn't simplify the equation... so it is correct to let x=cosθ? How is the second part related to the first part? Thanks in advance.

Look at the equation cos(4θ)-cos(3θ)=0 and expand the multiple angles in terms of cos(θ). Yes, put x=cos(θ). Then try to factor it.

 

[haruspex]

so it is correct to let x=cosθ?

Yes. Do you know how to expand cos(a+b)?