Cosine fun

Expanding $\cos{8\theta}$ into a function of $\cos{\theta}$ , we have:

$\begin{aligned} \cos{8\theta} & = 2\cos^2{4\theta} - 1 \\ & = 2\left( 2\cos^2{2\theta} - 1 \right)^2 - 1 \\ & = 2\left( 2\left(2\cos^2{\theta} - 1\right)^2 - 1 \right)^2 - 1 \\ & = 2\left( 2\left(4\cos^4{\theta} - 4\cos^2{\theta} + 1\right) - 1 \right)^2 - 1 \\ & = 2\left( 8\cos^4{\theta} - 8\cos^2{\theta} + 1 \right)^2 - 1 \\ & = 2\left( 64\cos^8{\theta} - 128\cos^6{\theta} + 80 \cos^4{\theta} -16\cos^2{\theta} + 1 \right) - 1 \\ & = 128\cos^8{\theta} - 256 \cos^6 {\theta} + 160 \cos^4{\theta} -32\cos^2{\theta} + 1 \end{aligned}$

$\Rightarrow$ When $\theta = 10^\circ\quad \Rightarrow x = \boxed{80}$ .

Use Chebyshev polynomials, specifically $T_{8}$ . This is basically the octuple angle formula, so 10 x 8 = 80 degrees. Or you could use algebraic manipulation using pythagorean identities and demoivre's theorem.