What is the value of `(cosec30^(@)-(1)/(sqrt(3)))` ?

Options

A. `((2sqrt(3)-1))/(sqrt(3))`
B. `((sqrt(3)-4))/(2sqrt(3))`
C. `(-1)/(sqrt(3))`
D. `(2)/(sqrt(3))`

Answer

To find the value of \( \csc 30^\circ - \frac{1}{\sqrt{3}} \), we will follow these steps:

### Step 1: Find the value of \( \csc 30^\circ \)
The cosecant function is the reciprocal of the sine function. Therefore, we need to find \( \csc 30^\circ \).

\[ \csc 30^\circ = \frac{1}{\sin 30^\circ} \]

We know that \( \sin 30^\circ = \frac{1}{2} \).

Thus,

\[ \csc 30^\circ = \frac{1}{\frac{1}{2}} = 2 \]

### Step 2: Substitute the value of \( \csc 30^\circ \) into the expression
Now we substitute \( \csc 30^\circ \) into the original expression:

\[ \csc 30^\circ - \frac{1}{\sqrt{3}} = 2 - \frac{1}{\sqrt{3}} \]

### Step 3: Simplify the expression
To simplify \( 2 - \frac{1}{\sqrt{3}} \), we need a common denominator. The common denominator here is \( \sqrt{3} \).

\[ 2 = \frac{2\sqrt{3}}{\sqrt{3}} \]

Now, we can rewrite the expression:

\[ 2 - \frac{1}{\sqrt{3}} = \frac{2\sqrt{3}}{\sqrt{3}} - \frac{1}{\sqrt{3}} = \frac{2\sqrt{3} - 1}{\sqrt{3}} \]

### Final Answer
Thus, the value of \( \csc 30^\circ - \frac{1}{\sqrt{3}} \) is:

\[ \frac{2\sqrt{3} - 1}{\sqrt{3}} \]