For the first problem, there are two competing meanings for the notation D_6. It is a dihedral group, with eith 6 elements or 12 elements, according to which definition you use. In either case, it can be described by two generators (a rotation and a reflection). This is a minimal set, because of you remove one generator then the remaining single generator could only generate an abelian group. But whichever way you define the group D_6, it is nonabelian.

I think that the second problem should read . The way to prove this is to put z=cos(β)+i.sin(β). Then, by de Moivre's theorem, . But .

Take the real part of those expressions to see that . But . Substitute that into the previous equation and you get the result.