Find the value of `cosec 10^(@) - sqrt(3) sec 10^(@)`.

Answer

To find the value of \( \csc 10^\circ - \sqrt{3} \sec 10^\circ \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \csc 10^\circ - \sqrt{3} \sec 10^\circ \]
We know that:
\[ \csc 10^\circ = \frac{1}{\sin 10^\circ} \quad \text{and} \quad \sec 10^\circ = \frac{1}{\cos 10^\circ} \]
So we can rewrite the expression as:
\[ \frac{1}{\sin 10^\circ} - \sqrt{3} \cdot \frac{1}{\cos 10^\circ} \]

### Step 2: Find a common denominator
To combine the fractions, we need a common denominator, which will be \( \sin 10^\circ \cos 10^\circ \):
\[ \frac{1 \cdot \cos 10^\circ}{\sin 10^\circ \cos 10^\circ} - \frac{\sqrt{3} \cdot \sin 10^\circ}{\sin 10^\circ \cos 10^\circ} \]
This simplifies to:
\[ \frac{\cos 10^\circ - \sqrt{3} \sin 10^\circ}{\sin 10^\circ \cos 10^\circ} \]

### Step 3: Use the double angle identity
We can use the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \) to rewrite the denominator:
\[ \sin 10^\circ \cos 10^\circ = \frac{1}{2} \sin 20^\circ \]
Thus, the expression becomes:
\[ \frac{\cos 10^\circ - \sqrt{3} \sin 10^\circ}{\frac{1}{2} \sin 20^\circ} = \frac{2(\cos 10^\circ - \sqrt{3} \sin 10^\circ)}{\sin 20^\circ} \]

### Step 4: Simplify the numerator
Now, we can factor the numerator. Notice that:
\[ \cos 10^\circ - \sqrt{3} \sin 10^\circ \]
can be expressed using the sine subtraction formula:
\[ \cos 10^\circ - \sqrt{3} \sin 10^\circ = 2 \left( \frac{1}{2} \cos 10^\circ - \frac{\sqrt{3}}{2} \sin 10^\circ \right) \]
Recognizing that \( \frac{1}{2} = \sin 30^\circ \) and \( \frac{\sqrt{3}}{2} = \cos 30^\circ \), we can write:
\[ \cos 10^\circ - \sqrt{3} \sin 10^\circ = 2 \sin(30^\circ - 10^\circ) = 2 \sin 20^\circ \]

### Step 5: Substitute back
Now substituting back into our expression:
\[ \frac{2(2 \sin 20^\circ)}{\sin 20^\circ} = \frac{4 \sin 20^\circ}{\sin 20^\circ} = 4 \]

### Final Answer
Thus, the value of \( \csc 10^\circ - \sqrt{3} \sec 10^\circ \) is:
\[ \boxed{4} \]