The value of $\mathrm{s}\mathrm{i}\mathrm{n}10°\mathrm{s}\mathrm{i}\mathrm{n}30°\mathrm{s}\mathrm{i}\mathrm{n}50°\mathrm{s}\mathrm{i}\mathrm{n}70°$ is:

$\sin 10°\sin 30°\sin 50°\sin 70°$

$=\sin 10°\times \frac{1}{2}\left(\frac{1}{2}\times 2\sin 50°\sin 70°\right)$

$=\frac{1}{4}\sin 10°\left(2\sin 50°\sin 70°\right)$

Using $2\sin A\sin B=\cos \left(A-B\right)-\cos \left(A+B\right)$

$=\frac{1}{4}\sin 10°\left(\cos 20°-\cos 120°\right)$

$=\frac{1}{4}\sin 10°\left(\cos 20°-\cos \left(90°+30°\right)\right)$

$=\frac{1}{4}\sin 10°\left(\cos 20°+\sin 30°\right)$

$=\frac{1}{4}\sin 10°\left(\cos 20°+\frac{1}{2}\right)$

$=\frac{1}{8}\left(2\sin 10°\cos 20°+\sin 10°\right)$

Again, using $2\sin A\cos B=\sin \left(A+B\right)-\sin \left(A-B\right)$ 

$=\frac{1}{8}\left(\sin 30°-\sin 10°+\sin 10°\right)$

$=\frac{1}{8}\left(\frac{1}{2}\right)=\frac{1}{16}.$