Find the value of `sqrt(3)cosec20^(@)-sec20^(@)`

Answer

To find the value of \( \sqrt{3} \csc 20^\circ - \sec 20^\circ \), we can follow these steps:

### Step 1: Rewrite the trigonometric functions
We know that:
\[ \csc 20^\circ = \frac{1}{\sin 20^\circ} \quad \text{and} \quad \sec 20^\circ = \frac{1}{\cos 20^\circ} \]
So we can rewrite the expression:
\[ \sqrt{3} \csc 20^\circ - \sec 20^\circ = \sqrt{3} \cdot \frac{1}{\sin 20^\circ} - \frac{1}{\cos 20^\circ} \]

### Step 2: Combine the terms
To combine the terms, we will find a common denominator, which is \( \sin 20^\circ \cos 20^\circ \):
\[ = \frac{\sqrt{3} \cos 20^\circ - \sin 20^\circ}{\sin 20^\circ \cos 20^\circ} \]

### Step 3: Use known values
We know that \( \sqrt{3} = \sin 60^\circ \) and \( \frac{1}{2} = \cos 60^\circ \). So we can express the numerator:
\[ = \frac{\sin 60^\circ \cos 20^\circ - \cos 60^\circ \sin 20^\circ}{\sin 20^\circ \cos 20^\circ} \]

### Step 4: Apply the sine subtraction formula
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 simplifies to:
\[ = \frac{\sin 40^\circ}{\sin 20^\circ \cos 20^\circ} \]

### Step 5: Use the double angle identity
We know that \( \sin 2\theta = 2 \sin \theta \cos \theta \). Therefore:
\[ \sin 40^\circ = 2 \sin 20^\circ \cos 20^\circ \]
Substituting this back into our expression gives:
\[ = \frac{2 \sin 20^\circ \cos 20^\circ}{\sin 20^\circ \cos 20^\circ} \]

### Step 6: Simplify the expression
The \( \sin 20^\circ \cos 20^\circ \) cancels out:
\[ = 2 \]

### Step 7: Multiply by 2
Since we factored out a 2 earlier, we multiply by 2:
\[ = 2 \cdot 2 = 4 \]

### Conclusion
Thus, the value of \( \sqrt{3} \csc 20^\circ - \sec 20^\circ \) is:
\[ \boxed{4} \]