2. $\frac{1}{\sin 10^{0}}$ $- 4\sin 70^{0}$
$=$ $\frac{1}{\sin 10^{0}}$ $- 4\sin \left ( 60^{0}+10^{0} \right )$
$=$ $\frac{1}{\sin 10^{0}}$ $- 4\left ( \sin 60^{0}\cos 10^{0} + \cos 60^{0}\sin 10^{0} \right )$
$=$ $\frac{1}{\sin 10^{0}}$ $- 4\left ( \frac{\sqrt{3}}{2}\cos 10^{0} + \frac{1}{2}\sin 10^{0} \right )$
$=$ $\frac{1}{\sin 10^{0}}$ $- 2\sqrt{3}\cos10^{0} - 2\sin10^{0}$
$=$ $\frac{1-2\sqrt{3}\sin 10^{0}\cos 10^{0}-2\sin ^{2}10^{0}}{\sin 10^{0}}$
$=$ $\frac{\cos 20^{0}-\sqrt{3}\sin 20^{0}}{\sin 10^{0}}$
$=$ $\frac{\cos 20^{0}-\frac{\sin 60^{0}}{\cos 60^{0}}\sin 20^{0}}{\sin 10^{0}}$
$=$ $\frac{\cos 20^{0}\cos 60^{0}-\sin 60^{0}\sin 20^{0}}{\sin 10^{0}}$
$=$ $\frac{\cos \left ( 60^{0}+20^{0} \right )}{\frac{1}{2}\sin 10^{0}}$
$=$ $\frac{\cos 80^{0}}{\frac{1}{2}\sin10^{0}}$
$=$ $\frac{\sin 10^{0}}{\frac{1}{2}\sin 10^{0}}$
$=$ $2$.