Question

$\frac{1}{\cos 80{}^{\circ }}-\frac{\sqrt{3}}{\sin 80{}^{\circ }}$ is equal to:

• A. $\sqrt{2}$
• B. $\sqrt{3}$
• C. 2
• D. 4
• E. $\sqrt{5}$

Answer 1

Answer: (D)

$\frac{\sin 80{}^{\circ }-\cos 80{}^{\circ }\sqrt{3}}{\sin 80{}^{\circ }\cos 80{}^{\circ }}$
$=\frac{2\left[\frac{1}{2}\sin 80{}^{\circ }-\frac{\sqrt{3}}{2}\cos 80{}^{\circ }\right]}{\sin 80{}^{\circ }\cos 80{}^{\circ }}$
$=\frac{4[\sin (80{}^{\circ }-60{}^{\circ })]}{2\sin 80{}^{\circ }\cos 80{}^{\circ }}$
$=\frac{4\sin 20{}^{\circ }}{2\sin 80{}^{\circ }}=\frac{4\sin 20{}^{\circ }}{\sin (\pi -20{}^{\circ })}=4$