The product $(1 - \omega + {\omega ^2})(1 - {\omega ^2} + {\omega ^4})(1 - {\omega ^4} + {\omega ^8}) \dots$ up to $2n$ factors is:

The value of $\frac{(\cos \theta + i \sin \theta)^4}{(\sin \theta + i \cos \theta)^5}$ is equal to,where $i = \sqrt{-1}$:

If $1, \omega, \omega^2$ are the cube roots of unity and $\alpha = \omega + 2\omega^2 - 3$,then $\alpha^3 + 12\alpha^2 + 48\alpha + 3$ equals

If $\omega$ is an imaginary cube root of unity, then the value of $(2-\omega)(2-\omega^{2}) + 2(3-\omega)(3-\omega^{2}) + \ldots + (n-1)(n-\omega)(n-\omega^{2})$ is

$\left(\frac{\sqrt{6}-\sqrt{2}}{4}+\frac{\sqrt{6}+\sqrt{2}}{4} i\right)^{2020} =$

Let $\omega$ be an imaginary cube root of unity. Then the value of $2(\omega + 1)(\omega^2 + 1) + 3(2\omega + 1)(2\omega^2 + 1) + \dots + (n + 1)(n\omega + 1)(n\omega^2 + 1)$ is