The sum of the products of the non-conjugate roots of $i^{1/4}$ taken two at a time is

Evaluate: $\frac{(\cos \theta + i\sin \theta)^4}{(\sin \theta + i\cos \theta)^5}$

$(\sin \theta + i\cos \theta )^n$ is equal to

If $\omega$ is a complex cube root of unity,then $(x+1)(x+\omega)(x-\omega-1)$ is equal to

If $\alpha, \beta \in \mathbb{C}$ are distinct roots of the equation $x^2 - x + 1 = 0$,then $\alpha^{101} + \beta^{107}$ is equal to:

For a non-$0$ complex number $z$,let $\arg (z)$ denote the principal argument of $z$,with $-\pi < \arg (z) \leq \pi$. Let $\omega$ be the cube root of unity for which $0 < \arg (\omega) < \pi$. Let $\alpha = \arg \left(\sum_{n=1}^{2025} (-\omega)^n\right)$. Then the value of $\frac{3 \alpha}{\pi}$ is $.....$ .