Complex Numbers Problems Part 5 ******************************** 201. Prove that .. math:: \left(\frac{1 + sin\phi + icos\phi}{1 + sin\phi - icos\phi}\right)^n = cos\left(\frac{n\pi}{2} + n\phi\right) + isin\left(\frac{n\pi}{2} - n\phi\right) 202. If :math:`sin\alpha + sin\beta + sin\gamma = cos\alpha + cos\beta + cos\gamma = 0,` show that :math:`cos3\alpha + cos3\beta + cos3\gamma = 3cos(\alpha + \beta + \gamma)` and :math:`sin3\alpha + sin3\beta + sin3\gamma = 3sin(\alpha + \beta + \gamma)` 203. If :math:`sin\alpha + sin\beta + sin\gamma = cos\alpha + cos\beta + cos\gamma = 0,` show that :math:`cos2\alpha + cos2\beta + cos2\gamma = sin2\alpha + sin2\beta + sin2\gamma = 0` 204. If :math:`\alpha, \beta` are the roots of the equation :math:`t^2 - 2t + 2 = 0,` show that a value of :math:`x,` satisfying .. math:: \frac{(x + \alpha)^n - (x + \beta)^b}{(\alpha - \beta)} = \frac{sin\theta}{sin^n\theta} \text{ is } x = cot\theta - 1 205. If :math:`(1 + x)^n = p_0 + p_1x + p_2x^2 + ... + p_nx^n,` show that .. math:: p_0 - p_2 + p_4 ... = 2^{\frac{n}{2}}cos\frac{n\pi}{4} \text{ and } p_1 - p_3 + p_5 + ... = 2^{\frac{n}{2}}sin\frac{n\pi}{4} 206. If :math:`(1 - x + x^2)^n = a_0 + a_1 + a_2x^2 + ... a_{2n}x^{2n}` show that .. math:: a_0 + a_3 + a_6 + ... = \frac{1}{3}\left(1 + 2^{n + 1}cos\frac{n\pi}{3}\right) 207. If :math:`n` is a positive integer and :math:`(1 + x)^n = c_0 + c_1x + c_2x^2 + ... + c_nx^n,` show that .. math:: c_0 + c_4 + c_8 + ... = 2^{n - 2} + 2^{\frac{n}{2} - 1}cos\frac{n\pi}{4}. 208. Solve the equation :math:`z^8 + 1 = 0` and deduce that .. math:: cos4\theta = 8\left(cos\theta - cos\frac{\pi}{8}\right)\left(cos\theta - cos\frac{3\pi}{8}\right)\left(cos\theta - cos\frac{5\pi}{8}\right)\left(cos\theta - cos\frac{7\pi}{8}\right) 209. Prove that the roots of the equation :math:`8x^3 - 4x^2 - 4x + 1 = 0` are :math:`cos\frac{\pi}{7}, cos\frac{3\pi}{7}, cos\frac{5\pi}{7}.` 210. Solve the equation :math:`z^{10} - 1 = 0` and deduce that .. math:: sin5\theta = 5sin\theta\left(1 - \frac{sin\theta}{sin^2\frac{\pi}{5}}\right)\left(1 - \frac{sin\theta}{sin^2\frac{2\pi}{5}}\right) 211. Solve the equation :math:`x^7 + 1 = 0` and deduce that .. math:: cos\frac{\pi}{7}cos\frac{3\pi}{7}cos\frac{5\pi}{7} = -\frac{1}{8} 212. Form the equation whose roots are :math:`cot^2\frac{\pi}{2n + 1}, cot^2\frac{2\pi}{2n + 1}, ..., cot^2\frac{n\pi}{2n + 1}` and hence find the value of :math:`cot^2\frac{\pi}{2n + 1} + cot^2\frac{2\pi}{2n + 1} + ... + cot^2\frac{n\pi}{2n + 1}` 213. If :math:`\theta \ne k\pi,` show that :math:`cos\theta sin\theta + cos^2\theta sin2\theta + ... +cos^n\theta sinn\theta = cot\theta(1 - cos^n\theta cosn\theta)` 214. Show that :math:`-3 -4i = 5e^{i\left(\pi + tan^{-1}\frac{4}{3}\right)}` 215. Solve the equation :math:`2\sqrt{2}x^4 = (\sqrt{3} - 1) + i(\sqrt{3} + 1)` 216. Show that :math:`\left(\frac{1 + cos\phi + isin\phi}{1 + cos\phi -isin\phi}\right)^n = cosn\phi + isinn\phi` 217. If :math:`2cos\theta = x + \frac{1}{x}` and :math:`2cos\phi = y + \frac{1}{y},` then prove that (i) :math:`\frac{x}{y} + \frac{y}{x} = 2cos(\theta - \phi)` (ii) :math:`xy + \frac{1}{xy} = 2cos(\theta + \phi)` (iii) :math:`x^my^n + \frac{1}{x^my^n} = 2cos(m\theta + n\phi)` (iv) :math:`\frac{x^m}{y^n} + \frac{y^n}{x^m} = 2cos(m\theta - n\phi)` 218. If :math:`\alpha, \beta` are the roots of the equation :math:`x^2 -2x +4 = 0,` prove that :math:`\alpha^n + \beta^n = 2^{n + 1}cos\frac{n\pi}{3}` 219. Find the equation whose roots are :math:`n` th powers of the roots of the equation :math:`x^2 -2xcos\theta + 1 = 0` 220. If :math:`\alpha, \beta` are imaginary cube roots of 1 then show that .. math:: \alpha e^{\alpha x} + \beta e^{\beta x} = -e^\frac{x}{2}\left[cos\left(\frac{\sqrt{3}}{2}x\right) + \sqrt{3}\left(\frac{\sqrt{3}}{2}x\right)\right] 221. Find the values of :math:`A` and :math:`B` where :math:`Ae^{2i\theta} + Be^{-2i\theta} = 5cos2\theta - 7sin2\theta` 222. If :math:`x = cos\theta + isin\theta` and :math:`\sqrt{1 - c^2} = nc - 1,` prove that .. math:: (1 + c cos\theta) = \frac{c}{2n}(1 + nx)\left(1 + \frac{n}{x}\right) 223. Show that the roots of equation :math:`(1 + z)^n = (1 -z)^n` are :math:`itan\frac{r\pi}{n}, r = 0, 1, 2, ..., (n - 1)` excluding the value when :math:`n` is even and :math:`r = \frac{n}{2}.` 224. If :math:`x = cos\alpha + isin\alpha, y = cos\beta + isin\beta,` show that .. math:: \frac{(x + y)(xy - 1)}{(x - y)(xy + 1)} = \frac{sin\alpha + sin\beta}{sin\alpha - sin\beta} Since we have not covered permutations and combinations let me give the formulas: .. math:: n! = 1 * 2 * 3 * ... * n 0! = 1 {n \choose r} = {^nCr} = C_r^n = \frac{n!}{r!(n-r)!} = \frac{n(n - 1) ... (n - r + 1)}{1 * 2 * 3 * ... * r } 225. Show that .. math:: ^nC_0 + {^nC_3} + {^nC_6} + ... = \frac{1}{3}\left[2^n + 2cos\frac{n\pi}{3}\right] 226. Show that .. math:: ^nC_1 + {^nC_4} + {^nC_7} + ... = \frac{1}{3}\left[2^{n - 2} + 2cos\frac{(n - 2)\pi}{3}\right] 227. Show that .. math:: ^nC_2 + {^nC_5} + {^nC_8} + ... = \frac{1}{3}\left[2^{n + 2} + 2cos\frac{(n + 2)\pi}{3}\right] 228. If :math:`(1 - x + x^2)^{6n} = a_0 + a_1x + a_2x^2 + ...,` show that .. math:: a_0 + a_3 + a_6 + ... = \frac{1}{3}(2^{6n + 1} + 1) 229. If :math:`(1 - x + x^2)^{n} = a_0 + a_1x + a_2x^2 + ...,` show that .. math:: a_0 + a_3 + a_6 + ... = \frac{1}{3}(1 + 2^{n + 1} cos\frac{n\pi}{3}) 230. Let .. math:: A = x + y +z, A' = x' + y' + z', AA' = x'' + y'' + z'', B = x + y\omega + z\omega^2, B' = x' + y'\omega + z'\omega^2, BB' = x'' + y''\omega + z''\omega^2, C = x + y\omega^2 + z\omega, C' = x' + y'\omega^2 + z'\omega, CC' = x'' + y''\omega^2 + z''\omega. then find :math:`x'', y''` and :math:`z''` in terms of :math:`x, y, z` and :math:`x', y', z'.` 231. Prove the equality .. math:: (ax - by -cz -dt)^2 + (bx + ay -dz + ct)^2 + (cx + dy + az -bt)^2 + (dx - cy + bz + at)^2 = .. math:: (a^2 + b^2 + c^2 + d^2)(x^2 + y^2 + z^2 + t^2). 232. Prove the following equalities: .. math:: \frac{cosn\theta}{cos^n\theta} = 1 - {^nC_2}tan^2\theta + {^nC_4}tan^4\theta - ... + A \text{ where } A = (-1)^\frac{n}{2}~tan^n\theta \text{ if } n \text{ is even,} A = (-1)^\frac{n - 1}{2}~{^nC_{n - 1}}tan^n\theta \text{ if } n \text{ is odd;} \frac{sinn\theta}{cos^n\theta} = {^nC_1}tan\theta - {^nC_3}tan^3\theta + {^nC_5}tan^5\theta - ... + A \text{ where } A = (-1)^\frac{n - 2}{2}~{^nC_{n - 1}}tan^{n - 1}\theta \text{ if } n \text{ is odd, } A = (-1)^\frac{n}{2}~tan^n\theta \text{ if } n \text{ is odd.} 233. Prove the following equality: .. math:: 2^{2m}cos^{2m}x = \sum_{k = 0}^{k = m - 1} 2 {2m \choose k} cos2(m - k)x + {2m \choose m} 234. Prove the following equality: .. math:: 2^{2m}sin^{2m}x = \sum_{k = 0}^{k = m - 1} (-1)^{m + k} 2 {2m \choose k} cos2(m - k)x + {2m \choose m} 235. Prove the following equality: .. math:: 2^{2m}cos^{2m + 1}x = \sum_{k = 0}^{k = m} 2 {{2m + 1} \choose k} cos(2m - 2k + 1)x 236. Prove the following equality: .. math:: 2^{2m}sin^{2m + 1}x = \sum_{k = 0}^{k = m} (-1)^{m + k} 2 {{2m + 1} \choose k} cos(2m - 2k + 1)x 237. Let .. math:: u_n = cos\alpha + r cos(\alpha + \theta) + r^2 cos(\alpha +2\theta) + ... + r^n cos(\alpha + n\theta) v_n = sin\alpha + r sin(\alpha + \theta) + r^2 sin(\alpha +2\theta) + ... + r^n sin(\alpha + n\theta) then show that .. math:: u_n = \frac{cos\alpha - r cos(\alpha - \theta) - r^{n + 1} cos[(n + 1)\theta + \alpha] + r^{n + 2} cos(n\theta + \alpha)}{1 - 2rcos\theta + r^2} v_n = \frac{sin\alpha - r sin(\alpha - \theta) - r^{n + 1} sin[(n + 1)\theta + \alpha] + r^{n + 2} sin(n\theta + \alpha)}{1 - 2rcos\theta + r^2} 238. Simplify the following sum: .. math:: S = 1 + n cos \theta + \frac{n(n - 1)}{1*2} cos2\theta + ... = \sum_{k = 0}^{k = n}C^n_k cosk\theta 239. Simplify the following sum: .. math:: S = 1 + n sin \theta + \frac{n(n - 1)}{1*2} sin2\theta + ... = \sum_{k = 0}^{k = n}C^n_k sink\theta 240. If :math:`\alpha = \frac{\pi}{2n}` and :math:`o < 2n` then prove that .. math:: sin^{2p} \alpha + sin^{2p} 2\alpha + ... + sin^{2p} n\alpha = \frac{1}{2} + n\frac{1 *3 * 5 * ... (2p - 1)}{2 * 4 * ... 2p} 241. Prove that the polynomial :math:`x(x^{n - 1} -na^{n - 1}) + a^n(n - 1)` is divisible by :math:`(x - a)^2.` 242. Prove that :math:`(x + y)^n - x^n - y^n` is divisible by :math:`xy(x + y)(x^2 + xy + y^2)` if :math:`n` is an odd number and not divisible by 3. 243. Find out whether the polynomial :math:`x^{4a} + x^{4b + 1} + x^{4c + 2} + x^{4d + 3}` is divisible by :math:`x^3 + x^2 + x + 1` where :math:`a, b, c, d` are positive integers. 244. Prove that the polynomial :math:`(cos\theta + x sin\theta)^n - \cos n\theta - x sin n\theta` is divisible by :math:`x^2 + 1.` 245. Prove that the polynomial :math:`x^n sin\theta - k^{n - 1}x sin n\theta + k^n sin(n - 1)\theta` is divisible by :math:`x^2 - 2kx cos\theta + k^2.` 246. Find the sum of the :math:`p` the powers of the roots of the equation :math:`x^n - 1 = 0` where :math:`p` is a positive integer. 247. Let :math:`\alpha = cos\frac{2\pi}{n} + isin\frac{2\pi}{n}` where :math:`n` is a positive integer and let .. math:: A_k = x + y\alpha^k + z\alpha^{2k} + ... + w\alpha^{(n - 1)k} \text{ where, } k = 0, 1, 2, 3 ..., n - 1 where, :math:`x, y, z, ..., u, w` and :math:`n` are arbitrary complex numbers. Prove that .. math:: \sum_{k = 0}^{k = n - 1}|A_k|^2 = n\{|x|^2 + |y|^2 + ... + |w|^2\} Prove the following identities: 248. .. math:: x^{2n} - 1= (x^2 - 1)\sum_{k = 1}^{k = n - 1}\left(x^2 - 2xcos\frac{k\pi}{n} + 1\right) 249. .. math:: x^{2n + 1} - 1 = (x - 1)\sum_{k = 1}^{k = n}\left(x^2 - 2xcos\frac{2k\pi}{2n + 1} + 1\right) 250. .. math:: x^{2n + 1} - 1= (x + 1)\sum_{k = 1}^{k = n}\left(x^2 + 2xcos\frac{2k\pi}{2n + 1} + 1\right)