Question

The value of $cosec{10}^{\circ }-\sqrt{3}sec{10}^{\circ }$ is

• A. $4$
• B. $2$
• C. $1$
• D. $\frac{1}{4}$

Answer 1

The correct option is A $4$

$cosec10°-\sqrt{3}\sec 10°$

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

$=\frac{\cos 10°-\sqrt{3}\sin 10°}{\sin 10°\cos 10°}$

$=2\left[\frac{\cos 10°-\sqrt{3}\sin 10°}{2\sin 10°\cos 10°}\right]$ [ Multiply and divide by $2$ ]

$=2\left[\frac{\cos 10°-\sqrt{3}\sin 10°}{\sin 20°}\right]$

$=2\times 2\left[\frac{\frac{1}{2}\cos 10°-\frac{\sqrt{3}}{2}\sin 10°}{\sin 20°}\right]$ [ Again multiply and divide by $2$ ]

$=4\left[\frac{\sin 30°\cos 10°-\cos 30°\sin 10°}{\sin 20°}\right]$

$=4\left[\frac{\sin (30°-10°)}{\sin 20°}\right]$ $[\therefore \sin (A-B)=\sin A\cos B-\cos A\sin B]$

$=4\left[\frac{\sin 20°}{\sin 20°}\right]$

$=4$

Hence, the answer is $4.$