If $\omega$ is a complex cube root of unity,then $\left(\frac{1-\sqrt{3} i}{2}\right)^{2020}+\left(\frac{1+\sqrt{3} i}{2}\right)^{2026} +\sin \left(\sum_{j=1}^6(j+\omega)(j+\omega^2) \frac{3 \pi}{152}\right)=$

If $e^{it} = \cos t + i \sin t$ and $e^{-it} = \cos t - i \sin t$, then $\cosh(x + iy) - \cosh(x - iy) =$

If $x=\frac{4}{5}+\frac{3}{5} i$ and $y=\frac{\sqrt{3}}{\sqrt{8}}-\frac{\sqrt{5}}{\sqrt{8}} i$,then $\left(x^2+\frac{1}{x^2}\right)\left(y^2-\frac{1}{y^2}\right)=$

Let $A = \{\theta \in [0, 2\pi] : 1 + 10 \operatorname{Re}\left(\frac{2 \cos \theta + i \sin \theta}{\cos \theta - 3i \sin \theta}\right) = 0\}$. Then $\sum_{\theta \in A} \theta^2$ is equal to

If $\alpha, \beta$ and $\gamma$ are angles that satisfy the following conditions, find the value of $xyz$.
$1.$ $\tan \alpha + \tan \beta + \tan \gamma = \tan \alpha \tan \beta \tan \gamma$
$2.$ $x = \cos \alpha + i \sin \alpha$
$3.$ $y = \cos \beta + i \sin \beta$
$4.$ $z = \cos \gamma + i \sin \gamma$

If $z = x + iy$ $(x, y \in R, x \neq -1/2)$,the number of values of $z$ satisfying $|z|^n = z^2|z|^{n-2} + z|z|^{n-2} + 1$ $(n \in N, n > 1)$ is