De Moivre's Theorem

If $x+\frac{1}{x}=\sqrt{2}$ then $x^{29}+\frac{1}{x^{29}}=$

If $\left|z\right|=1 \text{and} z\ne -1,$ then ${\left(\frac{1+z}{1+\overline{z}}\right)}^4+ {\left(\frac{1+\overline{z}}{1+z}\right)}^4$ is

Find the value of the complex numbers $(-1+i{)}^9$.

Using de Moivre's theorem, show that $\tan 4\theta$ can be written in the form $\frac{4\tan \theta -4{\tan }^3\theta }{1-6{\tan }^2\theta +{\tan }^4\theta }.$ Hence, solve the equation $7t^4+20t^3-42t^2-20t=-7,$ giving your answers to $3$ decimal places.

Determine the value of $\sum _{n=1}^{\infty }{\left(\frac{2}{3}\right)}^n\cos \left(\frac{n\pi }{3}\right),$ giving your answer in an exact form.

Find ${\cos }^8\theta +{\sin }^8\theta$ in the form $a\cos b\theta +c\cos d\theta +e$.

Simplify ${\cos }^5\theta +i{\sin }^5\theta$, giving your answer in an exponential form, where the terms are of the form $a{e}^{ki\theta }.$

Find ${\cos }^3\theta {\sin }^3\theta$ in terms of $\sin 6\theta$ and $\sin 2\theta .$

Find $4{\cos }^5\theta$ in terms of cosines of multiple angles.

If $Z+\frac{1}{Z}=2\sin \left(6°\right)$, then the value of $Z^{1000}+\frac{1}{Z^{1000}}$ is equal to

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

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 $i=\sqrt{-1}$)

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

Let $Z$ be a complex number satisfying the relation $Z^3+\frac{4{\left(\overline{Z}\right)}^2}{\left|Z\right|}=0$. If the least possible argument of $Z$ is $–k\pi$, then $k$ is equal to (here, $\arg Z\in \left(-\pi ,\pi \right]$)

The number of solutions of the equation $z^3+\frac{3{\left(\overline{z}\right)}^2}{\left|z\right|}=0$ (where, $z$ is a complex number) are

Cotyledons are also called-

Let $z=\cos \theta +i\sin \theta .$ Then the value of $\sum _{m=1}^{15}\mathrm{Im}\left(z^{2m-1}\right)$ at $\theta =2°$ is