The value of `(1)/( sin10^(@)) - ( sqrt( 3))/( cos 10^(@))=`

Options

A. 1
B. 4
C. 2
D. 0

Answer

To solve the expression \(\frac{1}{\sin 10^\circ} - \frac{\sqrt{3}}{\cos 10^\circ}\), we can follow these steps:

### Step 1: Write the expression
We start with the given expression:
\[ \frac{1}{\sin 10^\circ} - \frac{\sqrt{3}}{\cos 10^\circ} \]

### Step 2: Multiply by 2
To simplify the expression, we can multiply and divide by 2:
\[ \frac{1 \cdot 2}{\sin 10^\circ \cdot 2} - \frac{\sqrt{3} \cdot 2}{\cos 10^\circ \cdot 2} = \frac{2}{2\sin 10^\circ} - \frac{2\sqrt{3}}{2\cos 10^\circ} \]
This gives us:
\[ \frac{2}{2\sin 10^\circ} - \frac{2\sqrt{3}}{2\cos 10^\circ} = \frac{2}{2\sin 10^\circ} - \frac{\sqrt{3}}{\cos 10^\circ} \]

### Step 3: Rewrite using sine and cosine values
We can express \(\frac{2}{2\sin 10^\circ}\) and \(\frac{\sqrt{3}}{\cos 10^\circ}\) in terms of sine and cosine values:
\[ \frac{2}{2\sin 10^\circ} = \frac{1}{\sin 10^\circ} \quad \text{and} \quad \frac{\sqrt{3}}{2} = \cos 30^\circ \]
Thus, we can rewrite the expression as:
\[ \frac{1}{\sin 10^\circ} - \cos 30^\circ \cdot \frac{1}{\cos 10^\circ} \]

### Step 4: Find a common denominator
Now we will find a common denominator, which is \(2\sin 10^\circ \cos 10^\circ\):
\[ \frac{1 \cdot \cos 10^\circ - \sqrt{3} \cdot \sin 10^\circ}{\sin 10^\circ \cos 10^\circ} \]

### Step 5: Use the sine subtraction formula
The numerator can be recognized as:
\[ \cos 10^\circ - \sqrt{3} \sin 10^\circ = \sin(30^\circ - 10^\circ) = \sin 20^\circ \]
Thus, we have:
\[ \frac{2 \sin 20^\circ}{\sin 10^\circ \cos 10^\circ} \]

### Step 6: Simplify further
Using the identity \(2 \sin A \cos A = \sin(2A)\), we can simplify:
\[ = \frac{2 \sin 20^\circ}{\frac{1}{2} \sin 20^\circ} = 4 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{4} \]