$\sin {10}^0 \sin {50}^0 \sin {60}^0 \sin {70}^0 = \mathrm{m}$ and $\tan {20}^0 \tan {40}^0 \tan {60}^0 \tan {80}^0 = \mathrm{n}$ then $\frac{\mathrm{n}}{\mathrm{m}}=$

$$\begin{gathered} \sin {10}^0 \sin {50}^0 \sin {60}^0 \sin {70}^0 = \mathrm{m} \\ \frac{\sqrt{3}}{2}\frac{1}{4}\sin {30}^0=\mathrm{m} \\ \mathrm{m}=\frac{\sqrt{3}}{16} \end{gathered}$$
$$\begin{gathered} \tan {20}^0 \tan {40}^0 \tan {60}^0 \tan {80}^0 = \mathrm{n} \\ \sqrt{3}\tan {60}^{\mathrm{o}}=\mathrm{n} \\ \mathrm{n}=3 \\ \frac{\mathrm{n}}{\mathrm{m}}=\frac{3}{{\displaystyle \frac{\sqrt{3}}{16}}}=16\sqrt{3} \end{gathered}$$