Question

Find the value of $\csc {10}^{\circ }-\sqrt{3}\sec {10}^{\circ }$

Answer 1

$cosec10-\sqrt{3}sec10=\frac{1}{sin10}-\frac{\sqrt{3}}{cos10}$

$=\frac{cos10-\sqrt{3}sin10}{sin10cos10}=2\left[\frac{cos10-\sqrt{3}sin10}{2sin10cos10}\right]$

$=4\left[\frac{\frac{1}{2}cos10-\frac{\sqrt{3}}{2}sin10}{sin20}\right]=4\left[\frac{sin(30-10)}{sin20}\right]$

$=4\frac{sin20}{sin20}=4$