Complex Numbers

Show that $$(x \, + \, y)^2 \, + \, (x\omega \, + \, y\omega^2)^2 \, + \, (x\omega^2 \, + \, y\omega)^2 = 6xy.$$

Show that $$(x \, + \, y)^2 \, + \, (x\omega \, + \, y\omega^2)^2 \, + \, (x\omega^2 \, + \, y\omega)^2 = 6xy.$$

If $$a = \cos \dfrac {2\pi}{7} + i\sin \dfrac {2\pi}{7}$$, then the quadratic equation whose roots are $$\alpha = a + a^{2} + a^{4}$$ and $$\beta = a^{3} + a^{5} + a^{6}$$, is
If  $$a+ib=\dfrac {x+iy}{x-iy}$$ prove that $$a^2+b^2 =1$$ and $$\dfrac {b}{a}=\dfrac {2xy}{x^2-y^2}$$
In the multiplicative group of $$n$$th roots of unity, the inverse of $${ \omega  }^{ k }$$ is $$(k< n)$$: