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

Options

A. 2
B. 4
C. `2* sin20^circ/sin40^circ`
D. `4* sin20^circ/sin40^circ`

Answer

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

### Step 1: Rewrite the expression
We start with the expression:
\[ \sqrt{3} \csc 20^\circ - \sec 20^\circ \]
We can rewrite the cosecant and secant functions in terms of sine and cosine:
\[ \csc 20^\circ = \frac{1}{\sin 20^\circ} \quad \text{and} \quad \sec 20^\circ = \frac{1}{\cos 20^\circ} \]
Thus, the expression becomes:
\[ \sqrt{3} \cdot \frac{1}{\sin 20^\circ} - \frac{1}{\cos 20^\circ} \]

### Step 2: Multiply by a common factor
To simplify the expression, we can multiply the entire expression by 2 and divide by 2:
\[ \frac{2\sqrt{3}}{2\sin 20^\circ} - \frac{2}{2\cos 20^\circ} \]
This gives us:
\[ \frac{2\sqrt{3}}{2\sin 20^\circ} - \frac{2}{2\cos 20^\circ} \]

### Step 3: Substitute known values
We know that:
\[ \sqrt{3} = \sin 60^\circ \quad \text{and} \quad \frac{1}{2} = \cos 60^\circ \]
Substituting these values into the expression:
\[ \frac{2 \cdot \sin 60^\circ}{2\sin 20^\circ} - \frac{2 \cdot \cos 60^\circ}{2\cos 20^\circ} \]
This simplifies to:
\[ \frac{\sin 60^\circ}{\sin 20^\circ} - \frac{\cos 60^\circ}{\cos 20^\circ} \]

### Step 4: Combine the fractions
Now, we can combine these two fractions:
\[ \frac{\sin 60^\circ \cos 20^\circ - \cos 60^\circ \sin 20^\circ}{\sin 20^\circ \cos 20^\circ} \]
Using the sine subtraction formula \( \sin(a - b) = \sin a \cos b - \cos a \sin b \):
\[ = \frac{\sin(60^\circ - 20^\circ)}{\sin 20^\circ \cos 20^\circ} \]
This results in:
\[ = \frac{\sin 40^\circ}{\sin 20^\circ \cos 20^\circ} \]

### Step 5: Apply the double angle identity
Using the double angle identity \( \sin 2\theta = 2 \sin \theta \cos \theta \):
\[ = \frac{\sin 40^\circ}{\frac{1}{2} \sin 40^\circ} = 2 \]

### Step 6: Final simplification
Thus, we have:
\[ = 4 \]

### Conclusion
The final answer is:
\[ \sqrt{3} \csc 20^\circ - \sec 20^\circ = 4 \]