27. Complex Numbers Problems Part 6

251. \[x^{2n} + 1= (x + 1)\sum_{k = 0}^{k = n - 1}\left(x^2 - 2xcos\frac{(2k + 1)\pi}{2n} + 1\right) \]
252. \[sin \frac{\pi}{2n} sin \frac{2\pi}{2n} ... sin \frac{(n - 1)\pi}{2n} = \frac{\sqrt{n}}{2^{n - 1}} \]

     if \(n\) is even.

253. \[cos \frac{2\pi}{2n + 1} cos \frac{4\pi}{2n + 1} .. cos \frac{2n\pi}{2n + 1} = \frac{(-1)^\frac{n}{2}}{2^n} \]

     if \(n\) is even.

254. Prove that if \(cos\alpha + i sin\alpha\) is the solution of the equation \(x^n + p_1x^{n - 1} + ... + p_n = 0\) then \(p_1 sin\alpha + p_2 sin2\alpha + ... + p_n sinn\alpha = 0\) where \(p_1, p_2, ..., p_n\) are real.

255. Prove that

     \[\sqrt[3]{cos \frac{2\pi}{7}} + \sqrt[3]{cos \frac{4\pi}{7}} + \sqrt[3]{cos \frac{8\pi}{7}} = \sqrt[3]{\frac{1}{2}(5 - 3\sqrt[3]{7})} \]

256. Prove that

     \[\sqrt[3]{cos \frac{2\pi}{9}} + \sqrt[3]{cos \frac{4\pi}{9}} + \sqrt[3]{cos \frac{8\pi}{9}} = \sqrt[3]{\frac{1}{2}(3\sqrt[3]{9} - 6)} \]