In `DeltaDEF` is right angled at E.If sec `D=(25)/(7)` ,then what is the value of cosec F?

Options

A. `(7)/(25)`
B. `(24)/(7)`
C. `(25)/(7)`
D. `(7)/(24)`

Answer

To solve the problem, we need to find the value of cosec F in triangle DEF, which is right-angled at E. We are given that sec D = 25/7.

### Step-by-Step Solution:

1. **Understanding the Triangle**:
We have triangle DEF with a right angle at E. Let's denote the angles as follows:
- Angle D = ∠D
- Angle E = ∠E (90 degrees)
- Angle F = ∠F

2. **Using the Given Information**:
We know that sec D = 25/7. The secant function is defined as:
\[ \sec D = \frac{\text{Hypotenuse}}{\text{Base}} \]
In our triangle, we can denote:
- Hypotenuse (DF) = 25
- Base (DE) = 7

3. **Applying the Pythagorean Theorem**:
To find the length of EF (the height), we can use the Pythagorean theorem:
\[ DE^2 + EF^2 = DF^2 \]
Substituting the known values:
\[ 7^2 + EF^2 = 25^2 \]
This simplifies to:
\[ 49 + EF^2 = 625 \]

4. **Solving for EF**:
Rearranging the equation gives:
\[ EF^2 = 625 - 49 \]
\[ EF^2 = 576 \]
Taking the square root:
\[ EF = \sqrt{576} = 24 \]

5. **Finding Cosec F**:
The cosecant function is defined as:
\[ \csc F = \frac{\text{Hypotenuse}}{\text{Height}} = \frac{DF}{EF} \]
Substituting the values we found:
\[ \csc F = \frac{25}{24} \]

### Final Answer:
Thus, the value of cosec F is:
\[ \csc F = \frac{25}{24} \]