The value of `(1)/(sin 10^(@)) -(sqrt3)/(cos 10^(@))` is equal to

Options

A. 1
B. 2
C. 4
D. 3

Answer

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

### Step 1: Find a common denominator
The common denominator for the two fractions is \( \sin 10^\circ \cos 10^\circ \).

### Step 2: Rewrite the expression
Rewriting the expression with the common denominator gives:
\[ \frac{\cos 10^\circ}{\sin 10^\circ \cos 10^\circ} - \frac{\sqrt{3} \sin 10^\circ}{\sin 10^\circ \cos 10^\circ} \]

### Step 3: Combine the fractions
Now we can combine the fractions:
\[ \frac{\cos 10^\circ - \sqrt{3} \sin 10^\circ}{\sin 10^\circ \cos 10^\circ} \]

### Step 4: Simplify the numerator
The numerator is \( \cos 10^\circ - \sqrt{3} \sin 10^\circ \).

### Step 5: Recognize the form of the numerator
We can express the numerator in terms of sine and cosine of a compound angle. We know that:
\[ \cos A - \sin A = \sqrt{2} \cos(A + 45^\circ) \]
However, in this case, we can rewrite it directly as:
\[ \cos 10^\circ - \sin 10^\circ \cdot \sqrt{3} = 2 \left( \frac{1}{2} \cos 10^\circ - \frac{\sqrt{3}}{2} \sin 10^\circ \right) \]
This can be recognized as:
\[ 2 \left( \cos 30^\circ \cos 10^\circ - \sin 30^\circ \sin 10^\circ \right) = 2 \cos(30^\circ + 10^\circ) = 2 \cos 40^\circ \]

### Step 6: Substitute back into the expression
Now substituting this back into our expression gives:
\[ \frac{2 \cos 40^\circ}{\sin 10^\circ \cos 10^\circ} \]

### Step 7: Use the double angle identity
We know that \( \sin 2\theta = 2 \sin \theta \cos \theta \), so:
\[ \sin 20^\circ = 2 \sin 10^\circ \cos 10^\circ \]
Thus, we can rewrite our expression as:
\[ \frac{2 \cos 40^\circ}{\frac{1}{2} \sin 20^\circ} = \frac{4 \cos 40^\circ}{\sin 20^\circ} \]

### Step 8: Final simplification
Using the identity \( \sin(90^\circ - x) = \cos x \):
\[ \sin 20^\circ = \cos 70^\circ \]
Thus, the expression simplifies to:
\[ 4 \cdot \frac{\cos 40^\circ}{\sin 20^\circ} = 4 \]

### Final Answer
The value of \( \frac{1}{\sin 10^\circ} - \frac{\sqrt{3}}{\cos 10^\circ} \) is \( 4 \).

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