Tan inverse summation

Question

$$S=\sum\limits_{i=1}^{4}\tan^{-1} x_i$$

How to simplify this ?

I think I will have to use this :

but it looks too long a method .

Is there a method or symmetrical way which yields the answer quickly ?

note : $x_i$ are the roots of a fourth degree polynomial so I know the sum and product of the roots

Answer 1

$S$ is an argument of the complex number $(1+ix_1)(1+ix_2)(1+ix_3)(1+ix_4)$, and if you expand this, you can use what you know about $\sum x_i$ and $\prod x_i$ to simplify it a little.