If $z = x + iy$ and $x^2 + y^2 = 1$,then $\frac{1 + x + iy}{1 + x - iy} = $

$\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right) \times \left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right) \times \left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right) \times \ldots \infty =$

The value of $\left(\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right)^3$ is

If $z = x + iy$ satisfies $|z|-2=0$ and $|z-i|-|z+5i|=0$,then

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 $z_1$ and $z_2$ be any two non-zero complex numbers such that $3|z_1| = 4|z_2|$. If $z = \frac{3z_1}{2z_2} + \frac{2z_2}{3z_1}$,then: