`triangleXYZ` is right angled at Y. If `angleZ = 60^(@)` , then find the value of (cotX - `1/3`).

Options

A. `(3sqrt(3)-1)//3`
B. `(2sqrt(3)-sqrt(6))//2sqrt(3)`
C. `-5//3`
D. `(2-sqrt(3))//2sqrt(3)`

Answer

To solve the problem, we need to find the value of \( \cot X - \frac{1}{3} \) in triangle XYZ, which is a right triangle with a right angle at Y and angle Z equal to \( 60^\circ \).

### Step-by-Step Solution:

1. **Identify the Angles in the Triangle:**
Since triangle XYZ is a right triangle with the right angle at Y, we have:
\[ \angle Y = 90^\circ \]
Given that \( \angle Z = 60^\circ \), we can find \( \angle X \) using the fact that the sum of angles in a triangle is \( 180^\circ \):
\[ \angle X + \angle Y + \angle Z = 180^\circ \]
Substituting the known values:
\[ \angle X + 90^\circ + 60^\circ = 180^\circ \]
Simplifying this gives:
\[ \angle X + 150^\circ = 180^\circ \]
Therefore:
\[ \angle X = 180^\circ - 150^\circ = 30^\circ \]

2. **Calculate \( \cot X \):**
We know that:
\[ \cot X = \cot 30^\circ \]
The value of \( \cot 30^\circ \) is:
\[ \cot 30^\circ = \frac{1}{\tan 30^\circ} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \]

3. **Subtract \( \frac{1}{3} \) from \( \cot X \):**
Now we need to find \( \cot X - \frac{1}{3} \):
\[ \cot X - \frac{1}{3} = \sqrt{3} - \frac{1}{3} \]

4. **Combine the Terms:**
To combine \( \sqrt{3} - \frac{1}{3} \), we can express \( \sqrt{3} \) with a common denominator:
\[ \sqrt{3} = \frac{3\sqrt{3}}{3} \]
Thus:
\[ \sqrt{3} - \frac{1}{3} = \frac{3\sqrt{3}}{3} - \frac{1}{3} = \frac{3\sqrt{3} - 1}{3} \]

### Final Answer:
The value of \( \cot X - \frac{1}{3} \) is:
\[ \frac{3\sqrt{3} - 1}{3} \]