Question

Prove that : $\frac{1}{\sin {10}^{\circ }}-\frac{\sqrt{3}}{\cos {10}^{\circ }}=4$

Answer 1

LHS :

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

$=2[\frac{1}{2sin{10}^o}-\frac{\sqrt{3}}{2cos{10}^o}]$

$=2[\frac{sin{30}^o}{sin{10}^o}-\frac{cos{30}^o}{cos{10}^o}]$

$=2\left[\frac{sin{30}^{o}cos{10}^o-cos{30}^{o}sin{10}^o}{sin{10}^{o}cos{10}^o}\right]$

$=2\left[\frac{sin{20}^o}{sin{10}^{o}cos{10}^o}\right]$

$=2\times 2\left[\frac{sin{10}^{o}cos{10}^o}{sin{10}^{o}cos{10}^o}\right]$

$=4$ = RHS