Sum of inverse tangents of the roots of a polynomial

While messing around with the roots of polynomials, I came up with a couple very interesting questions:

Part one: (easy)

Part two: (much harder)

It is important to mention here that k is an integer, and also (for those of you that might have missed it), the coefficients of every two terms in the polynomial are equal.

The proof to part two is not here because my it takes up more than one page, however the answer is:

-pi/4 , which is a very nice answer if you ask me.

Feel free to show me your proofs for part two, I would like to see other ways (perhaps easier ways) to solve this