`DeltaABC`is right angled at B . If `mangleC=45^(@)` , then find the value of `(cosecA-sqrt(3))` .

Options

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

Answer

To solve the problem, we need to find the value of \( \csc A - \sqrt{3} \) given that triangle \( \Delta ABC \) is right-angled at \( B \) and \( \angle C = 45^\circ \).

### Step-by-Step Solution:

1. **Identify the Angles in Triangle ABC**:
Since triangle \( ABC \) is right-angled at \( B \), we know:
\[ \angle A + \angle B + \angle C = 180^\circ \]
Given that \( \angle B = 90^\circ \) and \( \angle C = 45^\circ \), we can find \( \angle A \):
\[ \angle A + 90^\circ + 45^\circ = 180^\circ \]
\[ \angle A + 135^\circ = 180^\circ \]
\[ \angle A = 180^\circ - 135^\circ = 45^\circ \]

2. **Calculate \( \csc A \)**:
We know that \( \csc A \) is the cosecant of angle \( A \). Since \( \angle A = 45^\circ \):
\[ \csc A = \frac{1}{\sin A} \]
The sine of \( 45^\circ \) is \( \frac{\sqrt{2}}{2} \), so:
\[ \csc 45^\circ = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

3. **Substitute into the Expression**:
Now, we substitute \( \csc A \) into the expression \( \csc A - \sqrt{3} \):
\[ \csc A - \sqrt{3} = \sqrt{2} - \sqrt{3} \]

4. **Final Answer**:
Thus, the value of \( \csc A - \sqrt{3} \) is:
\[ \sqrt{2} - \sqrt{3} \]

### Conclusion:
The final answer is \( \sqrt{2} - \sqrt{3} \).