The real part of $(1-\cos \theta+i \sin \theta)^{-1}$ is

If $z=\cos 6^{\circ}+i \sin 6^{\circ}$,then $\sum_{n=1}^{20} \operatorname{Im}\left(z^{2 n-1}\right)=$

Let the complex number $z = x + iy$ be such that $\frac{2z - 3i}{2z + i}$ is purely imaginary. If $x + y^2 = 0$,then $y^4 + y^2 - y$ is equal to:

The least value of $|z|$ where $z$ is a complex number satisfying the inequality $\exp \left(\frac{(|z|+3)(|z|-1)}{|z|+1} \log _{ e } 2\right) \geq \log _{\sqrt{2}}|5 \sqrt{7}+9 i |$,where $i=\sqrt{-1}$,is equal to:

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

If $z$ is a complex number satisfying $|z^3+z^{-3}| \leq 2$,then the maximum possible value of $|z+z^{-1}|$ is