The value of `sqrt3"cosec"20^@ - sec 20^@` is equal to

Options

A. 2
B. 1
C. 4
D. `-4`

Answer

To solve the expression \( \sqrt{3} \csc 20^\circ - \sec 20^\circ \), we will follow these steps:

### Step-by-Step Solution:

1. **Write the expression**:
\[ \sqrt{3} \csc 20^\circ - \sec 20^\circ \]

2. **Use trigonometric identities**:
Recall that:
\[ \csc \theta = \frac{1}{\sin \theta} \quad \text{and} \quad \sec \theta = \frac{1}{\cos \theta} \]
Thus, we can rewrite the expression as:
\[ \sqrt{3} \cdot \frac{1}{\sin 20^\circ} - \frac{1}{\cos 20^\circ} \]

3. **Combine the terms**:
To combine these fractions, we need a common denominator, which is \( \sin 20^\circ \cos 20^\circ \):
\[ \frac{\sqrt{3} \cos 20^\circ - \sin 20^\circ}{\sin 20^\circ \cos 20^\circ} \]

4. **Multiply numerator and denominator by 4**:
To simplify, we multiply both the numerator and the denominator by 4:
\[ \frac{4(\sqrt{3} \cos 20^\circ - \sin 20^\circ)}{4 \sin 20^\circ \cos 20^\circ} \]

5. **Rewrite the denominator using a trigonometric identity**:
The denominator can be rewritten using the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \):
\[ 4 \sin 20^\circ \cos 20^\circ = 2 \sin 40^\circ \]

6. **Substituting back**:
Now we have:
\[ \frac{4(\sqrt{3} \cos 20^\circ - \sin 20^\circ)}{2 \sin 40^\circ} \]

7. **Use the sine subtraction formula**:
The numerator can be simplified using the sine subtraction formula:
\[ \sqrt{3} \cos 20^\circ - \sin 20^\circ = \sin(60^\circ) \cos(20^\circ) - \cos(60^\circ) \sin(20^\circ = \sin(60^\circ - 20^\circ) = \sin(40^\circ) \]

8. **Final simplification**:
Thus, we have:
\[ \frac{4 \sin 40^\circ}{2 \sin 40^\circ} = \frac{4}{2} = 2 \]

### Final Answer:
\[ \sqrt{3} \csc 20^\circ - \sec 20^\circ = 2 \]