$\frac{(\sin \frac{\pi}{8} + i \cos \frac{\pi}{8})^8}{(\sin \frac{\pi}{8} - i \cos \frac{\pi}{8})^8}$ is equal to

If $\omega$ is the complex cube root of unity,then the value of $\left(3+5 \omega+3 \omega^2\right)^2+\left(3+3 \omega+5 \omega^2\right)^2$ is:

What is the product of all the values of $(1-i)^{\frac{2}{5}}$?

If $1, z_1, z_2, \ldots, z_{n-1}$ are the $n$th roots of unity,then the value of $(1-z_1)(1-z_2) \ldots (1-z_{n-1})$ is equal to

If $1, a, a^2, \ldots, a^{n-1}$ are the $n$th roots of unity,then $\sum_{i=1}^{n-1} \frac{1}{2-a^i}$ is equal to

If $\alpha, \beta, \gamma$ are the cube roots of $p$ $(p < 0)$,then for any $x, y$ and $z$,$\frac{x\alpha + y\beta + z\gamma}{x\beta + y\gamma + z\alpha} = $