Question

Prove that $\sqrt{3}\csc {20}^o-\sec {20}^o=4$

• A. $\frac{1}{4}$
• B. 1
• C. 3
• D. 4

Answer 1

Answer: (D)

4

$\sqrt{3}\csc 20-\sec 20=\frac{\sqrt{3}}{\sin 20}-\frac{1}{\cos 20}$
$=\frac{\sqrt{3}\cos 20-\sin 20}{\sin 20\cos 20}=\frac{2(\frac{\sqrt{3}}{2}\cos 20-\frac{1}{2}\sin 20)}{\frac{1}{2}\left(2\sin 20\cos 20\right)}$
$=\frac{4(\sin 60\cos 20-\cos 60\sin 20)}{\sin 40}=\frac{4\sin 40}{\sin 40}=4$