\begin{array}{l}
\displaystyle\cos(\pi/17)\cos(2\pi/17)\cos(4\pi/17)\cos(8\pi/17)\cos(16\pi/17)\cos(15\pi/17)\cos(13\pi/17)\cos(9\pi/17)=
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\displaystyle=\frac{\sin(\pi/17)\cos(\pi/17)\cos(2\pi/17)\cos(4\pi/17)\cos(8\pi/17)\cos(16\pi/17)\cos(15\pi/17)\cos(13\pi/17)\cos(9\pi/17)}{\sin(\pi/17)}
\vspace{0.5em}\\
\displaystyle=\frac{\sin(2\pi/17)\cos(2\pi/17)\cos(4\pi/17)\cos(8\pi/17)\cos(16\pi/17)\cos(15\pi/17)\cos(13\pi/17)\cos(9\pi/17)}{2\sin(\pi/17)}
\vspace{0.5em}\\
\displaystyle=\frac{\sin(4\pi/17)\cos(4\pi/17)\cos(8\pi/17)\cos(16\pi/17)\cos(15\pi/17)\cos(13\pi/17)\cos(9\pi/17)}{4\sin(\pi/17)}
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\displaystyle=\frac{\sin(8\pi/17)\cos(8\pi/17)\cos(16\pi/17)\cos(15\pi/17)\cos(13\pi/17)\cos(9\pi/17)}{8\sin(\pi/17)}
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\displaystyle=\frac{\sin(16\pi/17)\cos(16\pi/17)\cos(15\pi/17)\cos(13\pi/17)\cos(9\pi/17)}{16\sin(\pi/17)}
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\displaystyle=\frac{\sin(32\pi/17)\cos(15\pi/17)\cos(13\pi/17)\cos(9\pi/17)}{32\sin(\pi/17)}
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\displaystyle=-\frac{\sin(15\pi/17)\cos(15\pi/17)\cos(13\pi/17)\cos(9\pi/17)}{32\sin(\pi/17)}
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\displaystyle=-\frac{\sin(30\pi/17)\cos(13\pi/17)\cos(9\pi/17)}{64\sin(\pi/17)}
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\displaystyle=\frac{\sin(13\pi/17)\cos(13\pi/17)\cos(9\pi/17)}{64\sin(\pi/17)}
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\displaystyle=\frac{\sin(26\pi/17)\cos(9\pi/17)}{128\sin(\pi/17)}
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\displaystyle=-\frac{\sin(9\pi/17)\cos(9\pi/17)}{128\sin(\pi/17)}
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\displaystyle=-\frac{\sin(18\pi/17)}{256\sin(\pi/17)}
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\displaystyle=\frac{\sin(\pi/17)}{256\sin(\pi/17)}
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\displaystyle=\frac{1}{256}
\end{array}