Product of the sines

Firstly, we will find the value of $sin 10 sin 50 sin 70$ . This is equal to $cos 80 cos 40 cos 20$ because of the identity $cos x=sin (90-x)$ .

Let the product of the three cosines be A. Then we compute $8A \times sin 20$ . $8 sin 20 cos 20 cos 40 cos 80=4 sin 40 cos 40 cos 80=2 sin 80 cos 80=sin 160$ , because $sin (2x)=2 sin x cos x$ . Thus A is $\frac{1}{8} \times \frac{sin 160}{sin 20}$ . As sin 160=sin (180-160)=sin 20, A is $\frac{1}{8}$ . Thus the expression is equal to 16, with the fact that sin 30=0.5.