If complex number $z_1, z_2$ and $0$ are the vertices of an equilateral triangle, then $z_1^2+z_2^2-z_1{z}_2$ is equal to

If $\alpha$ is the angle which each side a regular polygon of $n$ side subtends at its centre, then 

$1+\cos \alpha +\cos 2\alpha +\cos 3\alpha +\dots \dots \dots +\cos (n-1)\alpha$ is equal to

Let, $z=1-t+i\sqrt{\left(t^2+t+2\right)}$, where $t$ is a real parameter. The locus of $z$ in the Argand plane is