If $$z = x + iy$$ and $$\omega = \left( {1 - iz} \right)/\left( {z - i} \right),$$ then $$\,\left| \omega \right| = 1$$ implies that, in the complex plane,

Show that $$$16\cos \left( {{{2\pi } \over {15}}} \right)\cos \left( {{{4\pi } \over {15}}} \right)\cos \left( {{{8\pi } \over {15}}} \right)\cos \left( {{{16\pi } \over {15}}} \right) = 1$$$

If $$\tan \,A = \left( {1 - \cos B} \right)/\sin B,$$ then $$tan2A = tan\,B$$.

Find all solutions of $$4{\cos ^2}\,x\sin x - 2{\sin ^2}x = 3\sin x$$