Question

The value of \( \csc 10^\circ - \sqrt{3}\sec 10^\circ \) is:

• A. \(1\)
• B. \(2\)
• C. \(4\)
• D. None of these

Hint:

Expressions of the form \( \cos\theta - \sqrt{3}\sin\theta \) can be simplified using \[ \cos\theta - \sqrt{3}\sin\theta = 2\cos(\theta + 60^\circ). \]

Answer 1

The Correct Option is C

Step 1: Write the given expression in terms of sine and cosine: \[ \csc 10^\circ - \sqrt{3}\sec 10^\circ = \frac{1}{\sin 10^\circ} - \frac{\sqrt{3}}{\cos 10^\circ} \]
Step 2: Take LCM of the denominators: \[ = \frac{\cos 10^\circ - \sqrt{3}\sin 10^\circ}{\sin 10^\circ \cos 10^\circ} \]
Step 3: Use the identity \[ \cos \theta - \sqrt{3}\sin \theta = 2\cos(\theta + 60^\circ) \] \[ \Rightarrow \cos 10^\circ - \sqrt{3}\sin 10^\circ = 2\cos 70^\circ = 2\sin 20^\circ \]
Step 4: Simplify the denominator: \[ \sin 10^\circ \cos 10^\circ = \frac{1}{2}\sin 20^\circ \]
Step 5: Substitute: \[ = \frac{2\sin 20^\circ}{\frac{1}{2}\sin 20^\circ} = 4 \]