Find the value of: $\frac{1}{\sin 10^{\circ}} - \frac{\sqrt{3}}{\cos 10^{\circ}}$

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answer is A.

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Detailed Solution

Given expression is,

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

Suppose,

S = \frac{1}{\sin 10^{\circ}} - \frac{\sqrt{3}}{\cos 10^{\circ}}

When 2 is multiplied and divided we get,

\Rightarrow S = 2 (\frac{1}{2 \sin 10^{\circ}} - \frac{\sqrt{3}}{2 \cos 10^{\circ}})

\Rightarrow S = 2 (\frac{\sin 30^{\circ}}{\sin 10^{\circ}} - \frac{\cos 30^{\circ}}{\cos 10^{\circ}}) [since \sin 30^{\circ} = \frac{1}{2}, \cos 30^{\circ} = \frac{\sqrt{3}}{2}]

\Rightarrow S = 2 (\frac{\sin 30^{\circ} \cos 10^{\circ} - \cos 30^{\circ} \sin 10^{\circ}}{\sin 10^{\circ} \cos 10^{\circ}})

\Rightarrow S = 2 (\frac{\sin 20^{\circ}}{\sin 10^{\circ} \cos 10^{\circ}}) [since, \sin Acos B - \cos Asin B = \sin (A - B)]

\Rightarrow S = 2 \times 2 (\frac{\sin 10^{\circ} \cos 10^{\circ}}{\sin 10^{\circ} \cos 10^{\circ}}) [∵ \sin 2 θ = 2 \sin θ \cos θ]

\Rightarrow S = 4

Thus,

\frac{1}{\sin 10^{\circ}} - \frac{\sqrt{3}}{\cos 10^{\circ}} = 4

Hence, option 1 is correct.

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