Question

$\left(\frac{1}{\sin 10°}\right)-\left(\frac{\surd 3}{\cos 10°}\right)=$

• A. $1$
• B. $0$
• C. $2$
• D. $4$

Answer 1

The correct option is D

$4$

Explanation for the correct option:

Simplify using standard trigonometric identities

Let $S=\frac{1}{\sin {10}^o}-\frac{\sqrt{3}}{\cos {10}^o}$

Multiply and divide by $2$:

$\Rightarrow S=2\left(\frac{1}{2\sin {10}^o}-\frac{\sqrt{3}}{2\cos {10}^o}\right)$

$\Rightarrow S=2\left(\frac{\sin {30}^o}{\sin {10}^o}-\frac{\cos {30}^o}{\cos {10}^o}\right)$ $\left[\mathbf{\because }\mathit{s}\mathit{i}\mathit{n}\mathbf{30}\mathbf{°}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{,}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{30}\mathbf{°}\mathbf{=}\frac{\sqrt{\mathbf{3}}}{\mathbf{2}}\right]$

$\Rightarrow S=2\left(\frac{\sin {30}^o\cos {10}^o-\cos {30}^o\sin {10}^o}{\sin {10}^o\cos {10}^o}\right)$

$\Rightarrow S=2\left(\frac{\sin {20}^o}{\sin {10}^o\cos {10}^o}\right)$ $\left[\mathbf{\because }\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathit{A}\mathit{c}\mathit{o}\mathit{s}\mathit{B}\mathbf{-}\mathit{c}\mathit{o}\mathit{s}\mathit{A}\mathit{s}\mathit{i}\mathit{n}\mathit{B}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathit{A}\mathbf{-}\mathit{B}\mathbf{)}\right]$

$\Rightarrow S=2\times 2\left(\frac{\sin {10}^o\cos {10}^o}{\sin {10}^o\cos {10}^o}\right)$ $\left[\mathbf{\because }\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{2}\mathit{\theta }\mathbf{=}\mathbf{2}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\right]$

$\Rightarrow S=\mathbf{4}$

Hence, Option ‘D’ is Correct.