Prove that $\cos20°\cos40°\cos60°\cos80°=\frac{1}{16}.$

Question

After applying product to sum formulae, I got this far $$\frac{\Bigl(1+2\cos(20)\Bigr)\Bigl(\cos(140)+\cos(20)\Bigr)}{8}=\frac{1}{16}.$$ How can I proceed further?

Answer 1

If you just want a proof, then ponder this: \begin{align*} &8\sin 20^\circ\cos20^\circ\cos40^\circ\cos80^\circ \\ = \, &4\sin 40^\circ\cos40^\circ\cos80^\circ \\ = \, &2\sin 80^\circ\cos80^\circ \\ = \, &\sin160^\circ=\sin20^\circ. \end{align*}