Does anyone know how to prove this formula?

All suggestions, ideas are welcome.

Below you find my unsuccessfull attempt using complex numbers.

When you convert it to complex numbers the equality can be rewritten as

The LHS of this equation contains terms in , , , ..., .

Calculate the coefficient for each term.

Now we still have to prove that for -n<k<n the coefficient of equals 0 to conclude the proof. But I don't know how to do this.

Other proofs are also welcome.

Thanks!