\begin{array}{l}
\displaystyle\cos(3\pi/17)\cos(6\pi/17)\cos(12\pi/17)\cos(7\pi/17)\cos(14\pi/17)\cos(11\pi/17)\cos(5\pi/17)\cos(10\pi/17)=
\vspace{0.5em}\\
=\displaystyle\frac{\sin(3\pi/17)\cos(3\pi/17)\cos(6\pi/17)\cos(12\pi/17)\cos(7\pi/17)\cos(14\pi/17)\cos(11\pi/17)\cos(5\pi/17)\cos(10\pi/17)}{\sin(3\pi/17)}
\vspace{0.5em}\\
=\displaystyle\frac{\sin(6\pi/17)\cos(6\pi/17)\cos(12\pi/17)\cos(7\pi/17)\cos(14\pi/17)\cos(11\pi/17)\cos(5\pi/17)\cos(10\pi/17)}{2\sin(3\pi/17)}
\vspace{0.5em}\\
=\displaystyle\frac{\sin(12\pi/17)\cos(12\pi/17)\cos(7\pi/17)\cos(14\pi/17)\cos(11\pi/17)\cos(5\pi/17)\cos(10\pi/17)}{4\sin(3\pi/17)}
\vspace{0.5em}\\
=\displaystyle\frac{\sin(24\pi/17)\cos(7\pi/17)\cos(14\pi/17)\cos(11\pi/17)\cos(5\pi/17)\cos(10\pi/17)}{8\sin(3\pi/17)}
\vspace{0.5em}\\
=\displaystyle-\frac{\sin(7\pi/17)\cos(7\pi/17)\cos(14\pi/17)\cos(11\pi/17)\cos(5\pi/17)\cos(10\pi/17)}{8\sin(3\pi/17)}
\vspace{0.5em}\\
=\displaystyle-\frac{\sin(14\pi/17)\cos(14\pi/17)\cos(11\pi/17)\cos(5\pi/17)\cos(10\pi/17)}{16\sin(3\pi/17)}
\vspace{0.5em}\\
=\displaystyle-\frac{\sin(28\pi/17)\cos(11\pi/17)\cos(5\pi/17)\cos(10\pi/17)}{32\sin(3\pi/17)}
\vspace{0.5em}\\
=\displaystyle\frac{\sin(11\pi/17)\cos(11\pi/17)\cos(5\pi/17)\cos(10\pi/17)}{32\sin(3\pi/17)}
\vspace{0.5em}\\
=\displaystyle\frac{\sin(22\pi/17)\cos(5\pi/17)\cos(10\pi/17)}{64\sin(3\pi/17)}
\vspace{0.5em}\\
=\displaystyle-\frac{\sin(5\pi/17)\cos(5\pi/17)\cos(10\pi/17)}{64\sin(3\pi/17)}
\vspace{0.5em}\\
=\displaystyle-\frac{\sin(10\pi/17)\cos(10\pi/17)}{128\sin(3\pi/17)}
\vspace{0.5em}\\
=\displaystyle-\frac{\sin(20\pi/17)}{256\sin(3\pi/17)}
\vspace{0.5em}\\
=\displaystyle\frac{\sin(3\pi/17)}{256\sin(3\pi/17)}
\vspace{0.5em}\\
=\displaystyle\frac{1}{256}
\end{array}