Find the value of $\sqrt3\csc 20^o -\sec 20^o$

Question

Find the value of $$\sqrt3\csc 20^o -\sec 20^o.$$

How do you solve this? Please help. Don't even know how to start this.

Answer 1

\begin{eqnarray}\sqrt3 \csc 20^o -\sec 20^o&=& {\sqrt3\over \sin 20^o} -{1\over \cos 20^o}\\ &&\\ &=&{\sqrt3 \cos 20^o - \sin 20^o\over \cos 20^o\sin 20^o}\\ &&\\ &=&4{\cos 30^o \cos 20^o - \sin 20^o\sin 30^o \over 2\cos 20^o\sin 20^o}\\ && \\ &=&4{ \cos 50^o \over \sin 40^o}= 4\end{eqnarray}

Answer 2

Hint:

$$ \sqrt3 \csc x - \sec x = \frac{\sqrt3}{\sin x} - \frac{1}{\cos x}, $$

and $\displaystyle 20^o = \frac{\pi}{9}$ radians.