As a note, this question requires some familiarity with the classic book referenced in the title, "Introduction to the Theory of Numbers", by Hardy and Wright. I have the Sixth Edition. I'm trying to provide most of the information required to ask my question, but realistically the book will probably need to be at hand. I'm hoping the book is sufficiently well known that someone will be able to answer from the information I'm giving here.
Equation (5.8.2) gives the cyclotomic polynomial $$\frac{x^{17} - 1}{x - 1} = x^{16} + x^{15} + \ldots + 1 = 0$$
whose roots are $\epsilon_{k} = {\it e}(\frac{k}{17}) = {\text cos}\ k\alpha + {\it i}\ {\text sin}\ k\alpha$, with $\alpha = \frac{2\pi}{17}$, and the notation ${\it e}(\tau) =_{\text def} e^{2\pi\ i\ \tau}$, and with $k = 1,\ldots,16$ here.
Below on the same page, variables $x_1$ and $x_2$ are each defined as sums of the $\epsilon$'s, in such a way that it is immediately obvious that their sum is $\Sigma_{1}^{16}\epsilon_{k}$, which (?) should be equal to $-1$ directly by (5.8.2). That is, $$x_1 = \epsilon_1 + \epsilon_9 + \epsilon_{13} + \epsilon_{15} + \epsilon_{16} + \epsilon_{8} + \epsilon_{4} + \epsilon_2$$ and $x_2$ is the sum of the remaining $\epsilon$'s.
However, in the book this simple step's result ($x_1 + x_2 = -1$) is arrived at only after a series of algebraic steps involving trigonometric identities is undertaken, e.g. $\epsilon_{k} + \epsilon_{17-k} = 2\ {\text cos}k\ \alpha$, $$x_1 = 2({\text cos}\ \alpha + {\text cos}\ 8\alpha + {\text cos}\ 4\alpha + {\text cos}\ 2\alpha)$$, and $$x_2 = 2({\text cos}\ 3\alpha + {\text cos}\ 7\alpha + {\text cos}\ 5\alpha + {\text cos}\ 6\alpha)$$.
Then comes the equation my question has to do with. As written in my book, it reads $$x_1 + x_2 = 2\Sigma_{1}^{8}{\text cos}k\alpha = 2\Sigma_{1}^{16}\epsilon_k = -1$$.
I have actually 2 specific questions:
First, why is the '2' multiplier present on the second summation $\Sigma_{1}^{16}\epsilon_k$?
Second, why is all this effort being undertaken since $x_1 + x_2 = -1$ follows immediately from their definitions together with equation (5.8.2)?
(1) The roots $\epsilon_k$ come in conjugate pairs. However, your $2$ in front of $\sum \epsilon_k$ is incorrect.
(2) Well, Hardy and Wright don't define $x_2$ to be $-1-x_1$. But they do observe that $x_2$ contains all the $\epsilon_j$ not present in $x_1$. The point of that last equation is that $$\sum_{k=1}^{16} \epsilon_k = -1.$$ This follows most easily from the fact that the sum of all the $17$th roots of unity is $0$, and you've omitted $\epsilon_0 = 1$.