Question

$\left(\frac{1}{\cos 80°}-\frac{\sqrt{3}}{\sin 80°}\right)$ is equal to

• A. $\sqrt{2}$
• B. $\sqrt{3}$
• C. $2$
• D. $4$
• E. $\sqrt{5}$

Answer 1

The correct option is D

$4$

Explanation for the correct option:

Step 1. Solve the given expression by using trigonometric identities:

$\frac{1}{\cos 80°}-\frac{\sqrt{3}}{\sin 80°}$

$=\left(\frac{\sin 80°-\sqrt{3}\cos 80°}{(\sin 80°\cos 80°}\right)$

Step 2. Multiply and divide numerator by 2:

$=\frac{2\left({\displaystyle \frac{1}{2}}\sin 80°-{\displaystyle \frac{\sqrt{3}}{2}}\cos 80°\right)}{(\sin 80°\cos 80°)}$
$=\frac{2(\sin 80°\cos 60°-\cos 80°\sin 60°)}{(\sin 80°\cos 80°)}$ $\left[\mathbf{\because }\mathit{s}\mathit{i}\mathit{n}\mathbf{60}\mathbf{°}\mathbf{}\mathbf{=}\mathbf{}\frac{\sqrt{\mathbf{3}}}{\mathbf{2}}\mathbf{,}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{60}\mathbf{°}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{1}}{\mathbf{2}}\right]$
Step 3. Multiply and divide numerator and denominator by 2:

$=\frac{4\sin (80°-60°)}{2(\sin 80°\cos °80°)}$ $\left[\mathbf{\because }\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathit{A}\mathbf{}\mathbf{-}\mathbf{}\mathit{B}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathit{A}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{B}\mathbf{}\mathbf{-}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{A}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathit{B}\right]$
$=\frac{4\sin (80°-60°)}{\sin 160°}$ $\left[\mathbf{\because }\mathbf{2}\mathit{s}\mathit{i}\mathit{n}\mathit{A}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathit{A}\mathbf{}\mathbf{=}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{2}\mathit{A}\right]$

$=\frac{4\sin (80°-60°)}{\sin (180°-20°)}$

$=4\frac{\sin 20°}{\sin 20°}$ $\left[\mathbf{\because }\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathbf{180}\mathbf{°}\mathbf{}\mathbf{-}\mathbf{}\mathit{A}\mathbf{)}\mathbf{}\mathbf{=}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathit{A}\right]$

$\mathbf{=}\mathbf{}\mathbf{4}$

Hence, Option ‘D’ is Correct.