$\cosh (\alpha + i\beta ) - \cosh (\alpha - i\beta )$ is equal to

If $\frac{(2-i) x+(1+i)}{2+i}+\frac{(1-2 i) y+(1-i)}{1+2 i}=1-2 i$,then $2 x+4 y=$

If $x$ and $y$ are two positive real numbers such that $x+iy = \frac{13 \sqrt{-5+12i}}{(2-3i)(3+2i)}$,then $13y-26x=$

If $\left(\frac{\cos \theta+i \sin \theta}{\sin \theta+i \cos \theta}\right)^{2020}+\left(\frac{1+\cos \theta+i \sin \theta}{1-\cos \theta+i \sin \theta}\right)^{2021} = x+i y$,then the value of $x+y$ at $\theta=\frac{\pi}{2}$ is

Let $Z$ and $W$ be complex numbers such that $|Z| = |W|$,and $\text{arg } Z$ denotes the principal argument of $Z$.
Statement $1$: If $\text{arg } Z + \text{arg } W = \pi$,then $Z = -\overline{W}$.
Statement $2$: $|Z| = |W|$ implies $\text{arg } Z - \text{arg } \overline{W} = \pi$.

If $i=\sqrt{-1}$,then $\sum_{n=0}^{\infty}\left(\frac{i}{3}\right)^n=$