Question

Cosec 10°-√3 sec 10°

Answer 1

hope this answer will help you

Answer 2

Answer:

The value of $\csc 10 ^ { \circ } - \sqrt { 3 } \sec 10 ^ { \circ }$ will be equal to 4.

Solution:

Given,

$\csc 10 ^ { \circ } - \sqrt { 3 } \sec 10 ^ { \circ }$

We know that the reciprocal of cosec and sec will be equal to sin and cos respectively. Thus,  

$\begin{aligned} \csc 10 ^ { \circ } & - \sqrt { 3 } \sec 10 ^ { \circ } = \frac { 1 } { \sin 10 ^ { \circ } } - \frac { \sqrt { 3 } } { \cos 10 ^ { \circ } } \\\\ & = \frac { \cos 10 ^ { \circ } - \sqrt { 3 } \sin 10 ^ { \circ } } { \sin 10 ^ { \circ } \cos 10 ^ { \circ } } \end{aligned}$

Multiply and divide by 2, we get,

$\begin{array} { c } { = 2 \left( \frac { \cos 10 ^ { \circ } - \sqrt { 3 } \sin 10 ^ { \circ } } { 2 \sin 10 ^ { \circ } \cos 10 ^ { \circ } } \right) \quad [ \because \sin 2 A = 2 \sin A \cos A ] } \\\\ { = 2 \times 2 \left( \frac { \frac { 1 } { 2 } \cos 10 ^ { \circ } - \frac { \sqrt { 3 } } { 2 } \sin 10 ^ { \circ } } { \sin 20 ^ { \circ } } \right) } \end{array}$

Hence, $\sin 30 ^ { \circ } = \frac { 1 } { 2 } \text { and } \cos 30 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }$

$\begin{array} { c } { = 4 \left( \frac { \sin 30 ^ { \circ } \cos 10 ^ { \circ } - \cos 30 ^ { \circ } \sin 10 ^ { \circ } } { \sin 20 ^ { \circ } } \right) } \\\\ { [ \because \sin ( A - B ) = \sin A \cos B - \cos A \sin B ] } \end{array}$

$\begin{aligned} = 4 & \left( \frac { \sin \left( 30 ^ { \circ } - 10 ^ { \circ } \right) } { \sin 20 ^ { \circ } } \right) \\\\ = 4 & \left( \frac { \sin 20 ^ { \circ } } { \sin 20 ^ { \circ } } \right) \\\\ & = 4 \end{aligned}$

Thus, the value of $\csc 10 ^ { \circ } - \sqrt { 3 } \sec 10 ^ { \circ }$ will be equal to 4.