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## Homework Statement

Show that for all integers n [tex]\geq[/tex] 1,

cos(2x) + cos(4x) + ... + cos(2nx) = [tex]\frac{1}{2}[/tex] ([tex]\frac{sin((2n+1)x)}{sin(x)}[/tex]-1)

Use this to verify that

[tex]\sum_{n=1}^{\infty}(\int_{0}^{\pi}[/tex] x([tex]\pi[/tex]-x)cos(2nx)dx) =

[tex]\frac{-1}{2}\int_{0}^{\pi}[/tex] x([tex]\pi[/tex]-x)dx)

## Homework Equations

## The Attempt at a Solution

I proved the first part of this problem using induction, however I don't see how I can use that to verify the second part. Maybe I can bring the summation into the integral and get the sum of cos(2nx), but I still don't see how that would give me what I need to prove. Any suggestions?