Fraction in a fraction...

$\frac{\frac{1}{sin 10} + \frac{\sqrt{3}}{cos 10}}{cos 20} = \frac{\frac{cos 10 + \sqrt{3} sin 10}{sin 10 cos 10}}{cos 20} = \frac{2(\frac{1}{2} cos 10 + \frac{\sqrt{3}}{2} sin 10)}{sin 10 cos 10 cos 20}$

Using special angles, we have $\frac{2(sin 30 cos 10 + cos 30 sin 10)}{sin 10 cos 10 cos 20} = \frac{2sin (30+10)}{sin 10 cos 10 cos 20} = \frac{2 sin 40}{sin 10 cos 10 cos 20}$ .

Applying double angle formula twice, $\frac{2 sin 40}{sin 10 cos 10 cos 20} = \frac{4 sin 40}{2 sin 10 cos 10 cos 20} = \frac{4 sin 40}{sin 20 cos 20} = \frac{8 sin 40}{2 sin 20 cos 20} = \frac{8 sin 40}{sin 40} = \boxed{8}$ .

$\\ \frac{\frac{1}{\sin 10^o} + \frac{\sqrt{3}}{\cos 10^o}}{\cos 20^o} = \\\\\\ \frac{\cos 10^o + \sqrt{3} \cdot \sin 10^o}{\cos 10^o \cdot \sin 10^o} \cdot \frac{1}{\cos 20^o}= \\\\\\ \frac{2 \cdot ( \frac{1}{2} \cdot \cos 10^o + \frac{\sqrt{3}}{2} \cdot \sin 10^o)}{\frac{1}{2} \cdot (2 \cdot \cos 10^o \cdot \sin 10^o)} \cdot \frac{1}{\cos 20^o}= \\\\\\ \frac{4 \cdot ( \cos 60^o \cdot \cos 10^o + \sin 60^o \cdot \sin 10^o)}{\sin 20^o} \cdot \frac{1}{\cos 20^o} = \\\\\\ \frac{4 \cdot \cos 50^o}{\sin 20^o \cdot \cos 20^o} = \\\\\\ \frac{4 \cdot \cos 50^o}{\frac{1}{2} \cdot (2 \cdot \sin 20^o \cdot \cos 20^o)} = \\\\\\ \frac{8 \cdot \cos 50^o}{\sin 40^o} =$

Uma vez que $\cos 50^o = \sin 40^o$ - complementares - temos que:

$\frac{8 \cdot \cos 50^o}{\cos 50^o} = \\\\ \boxed{\boxed{8}}$