find the value of`sqrt3cosec20^(@)-sec20^(@)`

Answer

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

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

### Step 2: Find a common denominator
The common denominator for the two fractions is \( \sin 20^\circ \cos 20^\circ \). Therefore, we can rewrite the expression:
\[ = \frac{\sqrt{3} \cos 20^\circ - \sin 20^\circ}{\sin 20^\circ \cos 20^\circ} \]

### Step 3: Multiply and divide by 2
To simplify the numerator, we can multiply and divide by 2:
\[ = \frac{2(\sqrt{3} \cos 20^\circ - \sin 20^\circ)}{2 \sin 20^\circ \cos 20^\circ} \]

### Step 4: Use known trigonometric identities
We know that:
\[ \sqrt{3} \cos 20^\circ - \sin 20^\circ = 2 \left( \frac{\sqrt{3}}{2} \cos 20^\circ - \frac{1}{2} \sin 20^\circ \right) \]
This can be recognized as the sine of a difference:
\[ = 2 \sin(60^\circ - 20^\circ) \]
Thus, we have:
\[ = \frac{4 \sin(40^\circ)}{2 \sin 20^\circ \cos 20^\circ} \]

### Step 5: Simplify using double angle identity
We know that:
\[ \sin(2\theta) = 2 \sin \theta \cos \theta \]
So, \( 2 \sin 20^\circ \cos 20^\circ = \sin 40^\circ \). Therefore, we can simplify:
\[ = \frac{4 \sin 40^\circ}{\sin 40^\circ} \]

### Step 6: Final simplification
The \( \sin 40^\circ \) terms cancel out:
\[ = 4 \]

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