`DeltaXYZ` is right angled at Y .If `mangleX=60^(@)`, then find the value of (cot Z+2).

Options

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

Answer

To solve the problem, we need to find the value of \( \cot Z + 2 \) given that triangle \( \Delta XYZ \) is right-angled at \( Y \) and \( \angle X = 60^\circ \).

### Step-by-Step Solution:

1. **Identify the Angles in the Triangle:**
Since triangle \( \Delta XYZ \) is right-angled at \( Y \), we know that the sum of the angles in a triangle is \( 180^\circ \). Therefore, we can express the angles as:
\[ \angle X + \angle Y + \angle Z = 180^\circ \]
Given that \( \angle Y = 90^\circ \) and \( \angle X = 60^\circ \), we can find \( \angle Z \):
\[ 60^\circ + 90^\circ + \angle Z = 180^\circ \]
\[ \angle Z = 180^\circ - 150^\circ = 30^\circ \]

2. **Find the Value of \( \cot Z \):**
We need to find \( \cot Z \). Since we have determined that \( \angle Z = 30^\circ \):
\[ \cot Z = \cot 30^\circ \]
The value of \( \cot 30^\circ \) is known to be:
\[ \cot 30^\circ = \frac{1}{\tan 30^\circ} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \]

3. **Calculate \( \cot Z + 2 \):**
Now, we can calculate \( \cot Z + 2 \):
\[ \cot Z + 2 = \sqrt{3} + 2 \]

4. **Final Answer:**
Thus, the final answer is:
\[ \cot Z + 2 = \sqrt{3} + 2 \]