What is the value of `( 1//3- cot60^@)`?

Options

A. `(2-sqrt3)//2sqrt3`
B. `(sqrt2-1)//sqrt2`
C. `(1-sqrt3)//3`
D. `(sqrt3-4)//2sqrt3`

Answer

To solve the question `(1/3 - cot 60°)`, we can follow these steps:

### Step 1: Identify the value of cot 60°
The cotangent of an angle in a right triangle is the ratio of the adjacent side to the opposite side. For 60 degrees, we can use the known value:
- \( \cot 60° = \frac{1}{\sqrt{3}} \)

### Step 2: Substitute the value of cot 60° into the expression
Now, we substitute the value of cot 60° into the expression:
- \( 1/3 - \cot 60° = 1/3 - \frac{1}{\sqrt{3}} \)

### Step 3: Find a common denominator
To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 3 and \( \sqrt{3} \) is \( 3\sqrt{3} \). We will convert both fractions:
- Convert \( 1/3 \) to have the common denominator:
\[ \frac{1}{3} = \frac{\sqrt{3}}{3\sqrt{3}} \]
- Convert \( \frac{1}{\sqrt{3}} \) to have the common denominator:
\[ \frac{1}{\sqrt{3}} = \frac{3}{3\sqrt{3}} \]

### Step 4: Subtract the fractions
Now we can subtract the two fractions:
\[ \frac{\sqrt{3}}{3\sqrt{3}} - \frac{3}{3\sqrt{3}} = \frac{\sqrt{3} - 3}{3\sqrt{3}} \]

### Step 5: Final answer
Thus, the final answer is:
\[ \frac{\sqrt{3} - 3}{3\sqrt{3}} \]