Find the numerical value of $\sin 10^\circ \sin 50^\circ \sin 70^\circ$. Prerequisite
This problem is found in "Trigonometry" by I. M. Gelfand [in English].
It is asked in the section "Double the angle". So, assume that I know the sin/cos angle additions [i.e.: $\sin(A + B) = \sin A \cos B + \cos A \sin B$, etc.] as well as everything learned prior.
I've check other sources and they say to use Morrie's Law, however I have not actually learned it in the book.

Problem
Find the numerical value of $\sin 10^\circ \sin 50^\circ \sin 70^\circ$.
Hint: If the value of the given expression is $M$, find $M \cos 10^\circ$.


Here's the strategy that I would use.
Note that
$$\sin 10^\circ\sin 50^\circ\sin 70^\circ = \frac{\sin 10^\circ\sin 30^\circ\sin 50^\circ\sin 70^\circ}{\sin 30^\circ}$$
And that 
$$\sin 10^\circ\sin 30^\circ\sin 50^\circ\sin 70^\circ = \frac{\sin 10^\circ\sin 20^\circ\sin 30^\circ\sin 40^\circ \sin 50^\circ\sin 60^\circ\sin 70^\circ\sin 80^\circ}{\sin 20^\circ\sin 40^\circ\sin 60^\circ\sin 80^\circ}$$
See where this is going?
