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

To solve the problem, we need to evaluate \( \csc 10^\circ - \sqrt{3}\sec 10^\circ \).

Let's start by understanding the trigonometric functions involved: 

• The cosecant function is defined as \( \csc \theta = \frac{1}{\sin \theta} \).
• The secant function is defined as \( \sec \theta = \frac{1}{\cos \theta} \).

Thus, our expression becomes:

\( \csc 10^\circ - \sqrt{3}\sec 10^\circ = \frac{1}{\sin 10^\circ} - \sqrt{3} \times \frac{1}{\cos 10^\circ} \)

We can combine the terms over a common denominator:

\( = \frac{\cos 10^\circ - \sqrt{3} \sin 10^\circ}{\sin 10^\circ \cos 10^\circ} \)

Noting that \( \sin 10^\circ \) and \( \cos 10^\circ \) are both positive in the first quadrant, we can simplify the numerator \( \cos 10^\circ - \sqrt{3} \sin 10^\circ \) to see if it simplifies to zero.

We know that trigonometric identities sometimes yield standard angles. So let's check:

• Notice that the equation closely resembles elements found in certain angles or angle identities.

Unfortunately, solving directly based on known angles or simplification does not seem straightforward without more advanced trigonometric identities or computational aids.

Ultimately, without further simplification, it's essential to check the correctness of options:

None of the standard values for simplification yield an integer directly, thus the given answer is **None of these**.

The correct answer is:

\(4\)