The value of `sqrt3 cosec20^@-sec20^@` is equal to:

Options

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

Answer

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

### Step 1: Rewrite the trigonometric functions
We know that:
- \( \csc 20^\circ = \frac{1}{\sin 20^\circ} \)
- \( \sec 20^\circ = \frac{1}{\cos 20^\circ} \)

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
To combine the two fractions, we need a common denominator, which will be \( \sin 20^\circ \cos 20^\circ \):
\[ = \frac{\sqrt{3} \cos 20^\circ}{\sin 20^\circ \cos 20^\circ} - \frac{\sin 20^\circ}{\sin 20^\circ \cos 20^\circ} \]

### Step 3: Combine the fractions
Now we can combine the fractions:
\[ = \frac{\sqrt{3} \cos 20^\circ - \sin 20^\circ}{\sin 20^\circ \cos 20^\circ} \]

### Step 4: Use the sine subtraction formula
We can recognize that the numerator can be expressed using the sine subtraction formula:
\[ \sqrt{3} \cos 20^\circ - \sin 20^\circ = 2 \left( \sin 60^\circ \cos 20^\circ - \cos 60^\circ \sin 20^\circ \right) \]
This simplifies to:
\[ = 2 \sin(60^\circ - 20^\circ) = 2 \sin 40^\circ \]

### Step 5: Substitute back into the expression
Now substitute this back into our expression:
\[ = \frac{2 \sin 40^\circ}{\sin 20^\circ \cos 20^\circ} \]

### Step 6: Use the double angle identity
Using the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \), we have:
\[ \sin 40^\circ = 2 \sin 20^\circ \cos 20^\circ \]
Thus, we can replace \( \sin 40^\circ \):
\[ = \frac{2 \cdot 2 \sin 20^\circ \cos 20^\circ}{\sin 20^\circ \cos 20^\circ} \]

### Step 7: Simplify the expression
The \( \sin 20^\circ \cos 20^\circ \) terms cancel out:
\[ = 4 \]

### Final Answer
Thus, the value of \( \sqrt{3} \csc 20^\circ - \sec 20^\circ \) is \( 4 \).

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