What is the valueof `(sqrt()2//3-cosec60^(@))`?

Options

A. `(sqrt()6-6)//3sqrt()3`
B. `(2-2sqrt()3)//sqrt()3`
C. `(1-sqrt()6)//sqrt()2`
D. `(4-sqrt()3)//2sqrt()3`

Answer

To solve the expression \( \left( \frac{\sqrt{2}}{3} - \csc 60^\circ \right) \), we will follow these steps:

### Step 1: Find the value of \( \csc 60^\circ \)
The cosecant function is the reciprocal of the sine function. Therefore, we need to find \( \sin 60^\circ \) first.

\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]

Thus,

\[ \csc 60^\circ = \frac{1}{\sin 60^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]

### Step 2: Substitute \( \csc 60^\circ \) back into the expression
Now we substitute the value of \( \csc 60^\circ \) into the original expression:

\[ \frac{\sqrt{2}}{3} - \csc 60^\circ = \frac{\sqrt{2}}{3} - \frac{2}{\sqrt{3}} \]

### Step 3: Find a common denominator
To subtract these fractions, we need a common denominator. The common denominator for \( 3 \) and \( \sqrt{3} \) is \( 3\sqrt{3} \).

### Step 4: Rewrite each term with the common denominator
We rewrite each term:

\[ \frac{\sqrt{2}}{3} = \frac{\sqrt{2} \cdot \sqrt{3}}{3\sqrt{3}} = \frac{\sqrt{6}}{3\sqrt{3}} \]

\[ \frac{2}{\sqrt{3}} = \frac{2 \cdot 3}{3\sqrt{3}} = \frac{6}{3\sqrt{3}} \]

### Step 5: Subtract the fractions
Now we can subtract the two fractions:

\[ \frac{\sqrt{6}}{3\sqrt{3}} - \frac{6}{3\sqrt{3}} = \frac{\sqrt{6} - 6}{3\sqrt{3}} \]

### Final Answer
Thus, the value of the expression \( \left( \frac{\sqrt{2}}{3} - \csc 60^\circ \right) \) is:

\[ \frac{\sqrt{6} - 6}{3\sqrt{3}} \]

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