How to find the value of this without using a calculator ?
Using the formulae in sections 16 and 17 of this page PlanetMath you can simplify $\displaystyle \sin(10)\sin(30)\sin(50)\sin(70)$ to arrive at a value of 1/16.
Similarly you can get $\displaystyle \cos(10)\cos(30)\cos(50)\cos(70)=\frac{3}{16}$.
You also need to remember that $\displaystyle \sin(x)=\sin(180-x)$.
I'll get you started. I'll put in extra steps to try to be clear. I'll leave the denominator to you.
$\displaystyle \sin(10)\sin(30)\sin(50)\sin(70)$
$\displaystyle =\frac{1}{2}\sin(10)\sin(50)\sin(70)$
$\displaystyle =\frac{1}{2}\left(\frac{1}{2}(\cos(50-10)-cos(50+10))\right) \sin(70)$
$\displaystyle =\frac{1}{2}\left(\frac{1}{2}(\cos(40)-cos(60))\right) \sin(70)$
$\displaystyle =\frac{1}{2}\left(\frac{1}{2}(\cos(40)-\frac{1}{2}\right) \sin(70)$
$\displaystyle =\frac{1}{4}\cos(40)\sin(70)-\frac{1}{8} \sin(70)$
$\displaystyle =\frac{1}{8}(\sin(70+40)+\sin(70-40))-\frac{1}{8} \sin(70)$
$\displaystyle =\frac{1}{8}\sin(110)+\frac{1}{16}-\frac{1}{8} \sin(70)$
$\displaystyle =\frac{1}{16} $
The LaTeX is harder than the maths.