Complex Numbers Problems Part 6
*******************************
251. .. math::
       x^{2n} + 1= (x + 1)\sum_{k = 0}^{k = n - 1}\left(x^2
       - 2xcos\frac{(2k + 1)\pi}{2n} + 1\right)

252. .. math::
       sin \frac{\pi}{2n} sin \frac{2\pi}{2n} ... sin \frac{(n - 1)\pi}{2n} =
       \frac{\sqrt{n}}{2^{n - 1}}

     if :math:`n` is even.

253. .. math::
       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 :math:`n` is even.

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

255. Prove that

     .. math::
        \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

     .. math::
        \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)}