Cosine identity

Whenever I see a trigonometric identity, my first attempt is just to use complex numbers to make the problem purely algebraic. It often is not the simplest solution, but it often makes the problem just about evaluating a polynomial expression. Here's how it works in this case:

If you let $w = e^{\pi i/7}$ , then $w^7+1=0$ and you can use the identity $\cos z = \dfrac{e^{iz}+e^{-iz}}{2}$ to write the left-hand side as $\begin{aligned} \left(16\cos(2\pi/7) + 8\cos(\pi/7) - 1\right)^2 &= \left(16\left(\frac{w^2+w^{-2}}{2}\right) + 8\left(\frac{w+w^{-1}}{2}\right) - 1\right)^2 \\ &= \frac{1}{w^4}\left(8w^4+4w^3-w^2+4w+8\right)^2 \\ &= \frac{1}{w^4}\left(64w^8 + 64w^7+56w^5+161w^4+56w^3+64w+64\right) \\ &= \frac{1}{w^4}\left(64(w+1)(w^7+1) + 161w^4 + 56(w^5+w^3)\right) \\ &= 161 + 56\left(w+w^{-1}\right) \\ &= 161 + 56\left(2\cos(\pi/7)\right) \\ &= 161 + 112\cos(\pi/7) \end{aligned}$

Therefore the answer is $161 + 112 = \boxed{273}$