Question

Find the value of $\frac{1}{\sin {10}^{\circ }}-\frac{\sqrt{3}}{\cos {10}^{\circ }}$.

Answer 1

$\frac{1}{\sin {10}^{\circ }}-\frac{\sqrt{3}}{\cos {10}^{\circ }}=$$\frac{\cos {10}^{\circ }-\sqrt{3}\sin {10}^{\circ }}{\sin {10}^{\circ }\cos {10}^{\circ }}$

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

using $sina.cosb-cosa.sinb=sin\left(\frac{a-b}{2}\right)$

$=\frac{4\sin {20}^{\circ }}{\sin {20}^{\circ }}=4$