By expansion of \(\log(1-2x\cos\theta+x^2)=\log(1-xe^{i\theta})+\log(1-xe^{-i\theta})\) in powers of \(x\), shew that \[ 2\cos n\theta = (2\cos\theta)^n - \frac{n}{1!}(2\cos\theta)^{n-2} + \frac{n(n-3)}{2!}(2\cos\theta)^{n-4} - \dots \] where \(n\) is a positive integer, obtaining the general term, and when \(n\) is even, the last term. Form the cubic equation whose roots are \(\cos^2\frac{\pi}{9}, \cos^2\frac{4\pi}{9}, \cos^2\frac{7\pi}{9}\).