The product of sines and the product of cosines of (k\pi)/(2n+1) for 0<k<n+1

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.

Re: The product of sines and the product of cosines of (k\pi)/(2n+1) for 0<k<n+1

Quote:

Originally Posted by

**ymar** I have to prove the following identities:

The suggestion is to prove and use this:

for

In the identity , have you tried putting z=–1 and z=1?

Re: The product of sines and the product of cosines of (k\pi)/(2n+1) for 0<k<n+1

Quote:

Originally Posted by

**Opalg** In the identity

, have you tried putting z=–1 and z=1?

Thanks! I haven't. I only tried . I wish I were able to see such things -- I really shouldn't study maths. :(

Re: The product of sines and the product of cosines of (k\pi)/(2n+1) for 0<k<n+1

Quote:

Originally Posted by

**ymar** I only tried

. I wish I were able to see such things -- I really shouldn't study maths. :(

If you tried then you were obviously thinking along the right lines. Don't be discouraged! (Nod)