# 18. Complex Numbers Problems Part 5¶

1. Prove that

$\left(\frac{1 + sin\phi + icos\phi}{1 + sin\phi - icos\phi}\right)^n = cos\left(\frac{n\pi}{2} + n\phi\right) + isin\left(\frac{n\pi}{2} - n\phi\right)$
2. If $$sin\alpha + sin\beta + sin\gamma = cos\alpha + cos\beta + cos\gamma = 0,$$ show that $$cos3\alpha + cos3\beta + cos3\gamma = 3cos(\alpha + \beta + \gamma)$$ and $$sin3\alpha + sin3\beta + sin3\gamma = 3sin(\alpha + \beta + \gamma)$$

3. If $$sin\alpha + sin\beta + sin\gamma = cos\alpha + cos\beta + cos\gamma = 0,$$ show that $$cos2\alpha + cos2\beta + cos2\gamma = sin2\alpha + sin2\beta + sin2\gamma = 0$$

4. If $$\alpha, \beta$$ are the roots of the equation $$t^2 - 2t + 2 = 0,$$ show that a value of $$x,$$ satisfying

$\frac{(x + \alpha)^n - (x + \beta)^b}{(\alpha - \beta)} = \frac{sin\theta}{sin^n\theta} \text{ is } x = cot\theta - 1$
5. If $$(1 + x)^n = p_0 + p_1x + p_2x^2 + ... + p_nx^n,$$ show that

\begin{align}\begin{aligned}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}\end{aligned}\end{align}
6. If $$(1 - x + x^2)^n = a_0 + a_1 + a_2x^2 + ... a_{2n}x^{2n}$$ show that

$a_0 + a_3 + a_6 + ... = \frac{1}{3}\left(1 + 2^{n + 1}cos\frac{n\pi}{3}\right)$
7. If $$n$$ is a positive integer and $$(1 + x)^n = c_0 + c_1x + c_2x^2 + ... + c_nx^n,$$ show that

$c_0 + c_4 + c_8 + ... = 2^{n - 2} + 2^{\frac{n}{2} - 1}cos\frac{n\pi}{4}.$
8. Solve the equation $$z^8 + 1 = 0$$ and deduce that

$cos4\theta = 8\left(cos\theta - cos\frac{\pi}{8}\right)\left(cos\theta - cos\frac{3\pi}{8}\right)\left(cos\theta - cos\frac{5\pi}{8}\right)\left(cos\theta - cos\frac{7\pi}{8}\right)$
9. Prove that the roots of the equation $$8x^3 - 4x^2 - 4x + 1 = 0$$ are $$cos\frac{\pi}{7}, cos\frac{3\pi}{7}, cos\frac{5\pi}{7}.$$

10. Solve the equation $$z^{10} - 1 = 0$$ and deduce that

$sin5\theta = 5sin\theta\left(1 - \frac{sin\theta}{sin^2\frac{\pi}{5}}\right)\left(1 - \frac{sin\theta}{sin^2\frac{2\pi}{5}}\right)$
11. Solve the equation $$x^7 + 1 = 0$$ and deduce that

$cos\frac{\pi}{7}cos\frac{3\pi}{7}cos\frac{5\pi}{7} = -\frac{1}{8}$
12. Form the equation whose roots are $$cot^2\frac{\pi}{2n + 1}, cot^2\frac{2\pi}{2n + 1}, ..., cot^2\frac{n\pi}{2n + 1}$$ and hence find the value of $$cot^2\frac{\pi}{2n + 1} + cot^2\frac{2\pi}{2n + 1} + ... + cot^2\frac{n\pi}{2n + 1}$$

13. If $$\theta \ne k\pi,$$ show that $$cos\theta sin\theta + cos^2\theta sin2\theta + ... +cos^n\theta sinn\theta = cot\theta(1 - cos^n\theta cosn\theta)$$

14. Show that $$-3 -4i = 5e^{i\left(\pi + tan^{-1}\frac{4}{3}\right)}$$

15. Solve the equation $$2\sqrt{2}x^4 = (\sqrt{3} - 1) + i(\sqrt{3} + 1)$$

16. Show that $$\left(\frac{1 + cos\phi + isin\phi}{1 + cos\phi -isin\phi}\right)^n = cosn\phi + isinn\phi$$

17. If $$2cos\theta = x + \frac{1}{x}$$ and $$2cos\phi = y + \frac{1}{y},$$ then prove that

1. $$\frac{x}{y} + \frac{y}{x} = 2cos(\theta - \phi)$$

2. $$xy + \frac{1}{xy} = 2cos(\theta + \phi)$$

3. $$x^my^n + \frac{1}{x^my^n} = 2cos(m\theta + n\phi)$$

4. $$\frac{x^m}{y^n} + \frac{y^n}{x^m} = 2cos(m\theta - n\phi)$$

18. If $$\alpha, \beta$$ are the roots of the equation $$x^2 -2x +4 = 0,$$ prove that $$\alpha^n + \beta^n = 2^{n + 1}cos\frac{n\pi}{3}$$

19. Find the equation whose roots are $$n$$ th powers of the roots of the equation $$x^2 -2xcos\theta + 1 = 0$$

20. If $$\alpha, \beta$$ are imaginary cube roots of 1 then show that

$\alpha e^{\alpha x} + \beta e^{\beta x} = -e^\frac{x}{2}\left[cos\left(\frac{\sqrt{3}}{2}x\right) + \sqrt{3}\left(\frac{\sqrt{3}}{2}x\right)\right]$
21. Find the values of $$A$$ and $$B$$ where $$Ae^{2i\theta} + Be^{-2i\theta} = 5cos2\theta - 7sin2\theta$$

22. If $$x = cos\theta + isin\theta$$ and $$\sqrt{1 - c^2} = nc - 1,$$ prove that

$(1 + c cos\theta) = \frac{c}{2n}(1 + nx)\left(1 + \frac{n}{x}\right)$
23. Show that the roots of equation $$(1 + z)^n = (1 -z)^n$$ are $$itan\frac{r\pi}{n}, r = 0, 1, 2, ..., (n - 1)$$ excluding the value when $$n$$ is even and $$r = \frac{n}{2}.$$

24. If $$x = cos\alpha + isin\alpha, y = cos\beta + isin\beta,$$ show that

$\frac{(x + y)(xy - 1)}{(x - y)(xy + 1)} = \frac{sin\alpha + sin\beta}{sin\alpha - sin\beta}$

Since we have not covered permutations and combinations let me give the formulas:

\begin{align}\begin{aligned}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 }\end{aligned}\end{align}
1. Show that

$^nC_0 + {^nC_3} + {^nC_6} + ... = \frac{1}{3}\left[2^n + 2cos\frac{n\pi}{3}\right]$
2. Show that

$^nC_1 + {^nC_4} + {^nC_7} + ... = \frac{1}{3}\left[2^{n - 2} + 2cos\frac{(n - 2)\pi}{3}\right]$
3. Show that

$^nC_2 + {^nC_5} + {^nC_8} + ... = \frac{1}{3}\left[2^{n + 2} + 2cos\frac{(n + 2)\pi}{3}\right]$
4. If $$(1 - x + x^2)^{6n} = a_0 + a_1x + a_2x^2 + ...,$$ show that

$a_0 + a_3 + a_6 + ... = \frac{1}{3}(2^{6n + 1} + 1)$
5. If $$(1 - x + x^2)^{n} = a_0 + a_1x + a_2x^2 + ...,$$ show that

$a_0 + a_3 + a_6 + ... = \frac{1}{3}(1 + 2^{n + 1} cos\frac{n\pi}{3})$
6. Let

\begin{align}\begin{aligned}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.\end{aligned}\end{align}

then find $$x'', y''$$ and $$z''$$ in terms of $$x, y, z$$ and $$x', y', z'.$$

7. Prove the equality

$(ax - by -cz -dt)^2 + (bx + ay -dz + ct)^2 + (cx + dy + az -bt)^2 + (dx - cy + bz + at)^2 =$
$(a^2 + b^2 + c^2 + d^2)(x^2 + y^2 + z^2 + t^2).$
8. Prove the following equalities:

\begin{align}\begin{aligned}\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.}\end{aligned}\end{align}
9. Prove the following equality:

$2^{2m}cos^{2m}x = \sum_{k = 0}^{k = m - 1} 2 {2m \choose k} cos2(m - k)x + {2m \choose m}$
10. Prove the following equality:

$2^{2m}sin^{2m}x = \sum_{k = 0}^{k = m - 1} (-1)^{m + k} 2 {2m \choose k} cos2(m - k)x + {2m \choose m}$
11. Prove the following equality:

$2^{2m}cos^{2m + 1}x = \sum_{k = 0}^{k = m} 2 {{2m + 1} \choose k} cos(2m - 2k + 1)x$
12. Prove the following equality:

$2^{2m}sin^{2m + 1}x = \sum_{k = 0}^{k = m} (-1)^{m + k} 2 {{2m + 1} \choose k} cos(2m - 2k + 1)x$
13. Let

\begin{align}\begin{aligned}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)\end{aligned}\end{align}

then show that

\begin{align}\begin{aligned}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}\end{aligned}\end{align}
14. Simplify the following sum:

$S = 1 + n cos \theta + \frac{n(n - 1)}{1*2} cos2\theta + ... = \sum_{k = 0}^{k = n}C^n_k cosk\theta$
15. Simplify the following sum:

$S = 1 + n sin \theta + \frac{n(n - 1)}{1*2} sin2\theta + ... = \sum_{k = 0}^{k = n}C^n_k sink\theta$
16. If $$\alpha = \frac{\pi}{2n}$$ and $$o < 2n$$ then prove that

$sin^{2p} \alpha + sin^{2p} 2\alpha + ... + sin^{2p} n\alpha = \frac{1}{2} + n\frac{1 *3 * 5 * ... (2p - 1)}{2 * 4 * ... 2p}$
17. Prove that the polynomial $$x(x^{n - 1} -na^{n - 1}) + a^n(n - 1)$$ is divisible by $$(x - a)^2.$$

18. Prove that $$(x + y)^n - x^n - y^n$$ is divisible by $$xy(x + y)(x^2 + xy + y^2)$$ if $$n$$ is an odd number and not divisible by 3.

19. Find out whether the polynomial $$x^{4a} + x^{4b + 1} + x^{4c + 2} + x^{4d + 3}$$ is divisible by $$x^3 + x^2 + x + 1$$ where $$a, b, c, d$$ are positive integers.

20. Prove that the polynomial $$(cos\theta + x sin\theta)^n - \cos n\theta - x sin n\theta$$ is divisible by $$x^2 + 1.$$

21. Prove that the polynomial $$x^n sin\theta - k^{n - 1}x sin n\theta + k^n sin(n - 1)\theta$$ is divisible by $$x^2 - 2kx cos\theta + k^2.$$

22. Find the sum of the $$p$$ the powers of the roots of the equation $$x^n - 1 = 0$$ where $$p$$ is a positive integer.

23. Let $$\alpha = cos\frac{2\pi}{n} + isin\frac{2\pi}{n}$$ where $$n$$ is a positive integer and let

\begin{align}\begin{aligned}A_k = x + y\alpha^k + z\alpha^{2k} + ... + w\alpha^{(n - 1)k}\\\text{ where, } k = 0, 1, 2, 3 ..., n - 1\end{aligned}\end{align}

where, $$x, y, z, ..., u, w$$ and $$n$$ are arbitrary complex numbers.

Prove that

$\sum_{k = 0}^{k = n - 1}|A_k|^2 = n\{|x|^2 + |y|^2 + ... + |w|^2\}$

Prove the following identities:

1. $x^{2n} - 1= (x^2 - 1)\sum_{k = 1}^{k = n - 1}\left(x^2 - 2xcos\frac{k\pi}{n} + 1\right)$
2. $x^{2n + 1} - 1 = (x - 1)\sum_{k = 1}^{k = n}\left(x^2 - 2xcos\frac{2k\pi}{2n + 1} + 1\right)$
3. $x^{2n + 1} - 1= (x + 1)\sum_{k = 1}^{k = n}\left(x^2 + 2xcos\frac{2k\pi}{2n + 1} + 1\right)$