$\cos 20°.\cos 40°.\cos 60°.\cos 80°=$

$\cos 20°.\cos 40°.\cos 60°.\cos 80°$

$=\cos 40°\cos 80°.\frac{1}{2}\times \cos 20°$

$=\cos {(60°-20°)}^0\cos {(60°+20°)}^0\times \frac{\cos 20°}{2}$

$=\frac{1}{4}-{\sin }^{2}20°.\frac{\cos 20°}{2}$

$=\frac{1-4{\sin }^{2}20°}{4}.\frac{\cos 20°}{2}$

$=\frac{1-4+4{\cos }^{2}20°}{4}.\frac{\cos 20°}{2}$

$=\frac{-3+4{\cos }^{2}20°}{4}.\frac{\cos 20°}{2}$

$=\frac{4{\cos }^{3}20°-3\cos 20°}{8}$

$=\frac{\cos 3\left(20\right)}{8}$

$\therefore 4{\cos }^3\theta -3\cos \theta =\cos 3\theta$

$=\frac{\cos 60°}{8}$

$=\frac{1}{\frac{2}{8}}$

$=\frac{1}{16}$