De Moivre's Theorem

Find the value of $k$ if ${\left[\frac{1+\sin \frac{\pi }{8}+i\cos \frac{\pi }{8}}{1+\sin \frac{\pi }{8}-i\cos \frac{\pi }{8}}\right]}^{\frac{8}{3}}=-k$.

The value of $\sqrt[4]{-1}$ is 

Find the value of ${\left(\frac{1+\sin \frac{\pi }{10}+i\cos \frac{\pi }{10}}{1+\sin \frac{\pi }{10}-i\cos \frac{\pi }{10}}\right)}^{10}$.

If $z={\left(\frac{\sqrt{3}}{2}+\frac{i}{2}\right)}^5+{\left(\frac{\sqrt{3}}{2}-\frac{i}{2}\right)}^5+{\left(1+i\right)}^2+{\left(1-i\right)}^2$, then  (where $i=\sqrt{-1}$)

Consider a regular $10-$gon with its vertices on the unit circle. With one vertex fixed, draw straight lines to the other $9$ vertices. Call them ${\mathrm{L}}_1, {\mathrm{L}}_2,\dots ,{\mathrm{L}}_9$ and denote their lengths by $l_1,l_2,\dots ,l_9$ respectively. Then the product $l_1{l}_2\dots l_9$ is

If $n$ is an integer and $z=cis\theta$, then $\frac{z^{2n}-1}{z^{2n}+1}=$

Let $a=\cos 1°$ and $b=\sin 1°$. We say that a real number is algebraic if it is a root of a polynomial with integer coefficients. Then,

If $2\cos \frac{7\mathrm{\pi }}{5}$ is one of the values of $z^{\frac{1}{5}}$, then $\mathrm{z}=$

If the complex number $a$ is such that $\left|a\right|=1,$ and $\arg \left(a\right)=\theta$, then the roots of the equation ${\left(\frac{1+iz}{1-iz}\right)}^4=a$ are $z=$

${\left(\frac{1+\cos \frac{\pi }{8}-i\sin \frac{\pi }{8}}{1+\cos \frac{\pi }{8}+i\sin \frac{\pi }{8}}\right)}^{12}=$

If $a_1, a_{2 },\dots , a_n$  are real numbers with $a_n\ne 0$ and $\cos \alpha +i\sin \alpha$ is a root of $z^n+a_1{z}^{n-1}+a_2{z}^{n-2}+...+a_{n-1}z+a_n=0$, then the sum $a_1\cos \alpha +a_2\cos 2\alpha +a_3\cos 3\alpha +...+a_n\cos n\alpha$ is

The number of roots of equation $z^5=\overline{z}$ is

The real part of each root of the equation ${\left(z+1\right)}^6+z^6=0$ is: (where $z$ is a complex number)

If  $z+\frac{1}{z}=2\cos 5^o$, then the value of $z^{12} + \frac{1}{z^{12}} + 2017$ is equal to :(where $z$ is a complex number)

 What is the value of $2^{\left|1+\alpha +{\alpha }^2+{\alpha }^{-2}-{\alpha }^{-1}\right|}$  when $\alpha$ is non real and $\alpha =\sqrt[5]{1}$ :

The number of solution(s) of the equation $z^2=4z+{\left|z\right|}^2\mathrm{ }+\frac{16}{{\left|z\right|}^3}$ is (where $z=x+iy, x,y\in R,i^2=–1 \mathrm{a}\mathrm{n}\mathrm{d} x\ne 2$)

Given if $\mathrm{m}$ and $\mathrm{x}$ are two real numbers, then the real value of ${\mathrm{e}}^{2{\mathrm{micot}}^{-1}\mathrm{x}}{\left(\frac{\mathrm{xi}+1}{\mathrm{xi}-1}\right)}^{\mathrm{m}}$ is equal to

If $z_n=\cos \frac{\pi }{(2n+1)(2n+3)}+i\sin \frac{\pi }{(2n+1)(2n+3)},$
then find $Im\left(\underset{n\to \infty }{\lim }{z}_1{z}_2{z}_3\dots z_n\right).$

If $z=1+ai$ is a complex number, $a>0$, such that $z^3$ is a real number. Then the value of the sum $1+z+z^2+\dots .+z^{11}$ is equal to: