I have to prove the following identities:

The suggestion is to prove and use this:

for

I can prove this last one. The roots of the polynomial on the left-hand side are They are also the roots of the quadratic expressions on the right-hand side. I don't know how to use it though. The only way I have found to see and here is this:

Now I know that if and then for every and whether we take the plus or the minus. This is equations but I really have no idea what to do with them.

Of course, if you have any ways of solving this without this suggestion, I will appreciate them to.

Edit: z^n corrected to z^k in the third formula. Sorry.