`DeltaDEF` is right angled at E . If `mangleD=45^(@)` , then what is the value of `cosecFxxcotD` ?

Options

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

Answer

To solve the problem, we need to find the value of \( \csc F \times \cot D \) in triangle \( \Delta DEF \), which is right-angled at \( E \) with \( \angle D = 45^\circ \).

### Step-by-Step Solution:

1. **Identify the Angles in Triangle DEF**:
- Given that \( \Delta DEF \) is a right triangle with \( \angle E = 90^\circ \) and \( \angle D = 45^\circ \).
- Since the sum of angles in a triangle is \( 180^\circ \), we can find \( \angle F \):
\[ \angle F = 180^\circ - \angle D - \angle E = 180^\circ - 45^\circ - 90^\circ = 45^\circ \]

2. **Calculate \( \csc F \)**:
- The cosecant function is the reciprocal of the sine function:
\[ \csc F = \frac{1}{\sin F} \]
- Since \( \angle F = 45^\circ \):
\[ \csc 45^\circ = \frac{1}{\sin 45^\circ} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \]

3. **Calculate \( \cot D \)**:
- The cotangent function is the reciprocal of the tangent function:
\[ \cot D = \frac{1}{\tan D} \]
- Since \( \angle D = 45^\circ \):
\[ \cot 45^\circ = \frac{1}{\tan 45^\circ} = \frac{1}{1} = 1 \]

4. **Combine the Results**:
- Now we can find \( \csc F \times \cot D \):
\[ \csc F \times \cot D = \sqrt{2} \times 1 = \sqrt{2} \]

### Final Answer:
The value of \( \csc F \times \cot D \) is \( \sqrt{2} \).

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