`DeltaDEF` is right angled at E.If `cosecD=(25)/(24)`, then what is the value of cos F ?

Options

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

Answer

To solve the problem, we need to find the value of cos F in the right-angled triangle DEF, where E is the right angle and we are given that cosec D = 25/24.

### Step-by-Step Solution:

1. **Understand the Triangle and Given Information**:
- We have a right-angled triangle DEF with the right angle at E.
- We know that cosec D = 25/24.

2. **Relate Cosecant to Sides**:
- Cosecant (cosec) is the reciprocal of sine. Thus, cosec D = 1/sin D.
- Therefore, sin D = 24/25 (since cosec D = 25/24).

3. **Identify the Sides of the Triangle**:
- In triangle DEF, let:
- The hypotenuse (DF) = 25 (since sin D = opposite/hypotenuse).
- The opposite side to angle D (EF) = 24 (since sin D = 24/25).

4. **Use Pythagoras Theorem to Find the Base**:
- We need to find the length of the base (DE).
- According to the Pythagorean theorem:
\[ DF^2 = DE^2 + EF^2 \]
- Substituting the known values:
\[ 25^2 = DE^2 + 24^2 \]
\[ 625 = DE^2 + 576 \]
- Rearranging gives:
\[ DE^2 = 625 - 576 = 49 \]
- Taking the square root:
\[ DE = 7 \]

5. **Find cos F**:
- Cosine of angle F is defined as the adjacent side over the hypotenuse:
\[ \cos F = \frac{DE}{DF} \]
- Substituting the values we found:
\[ \cos F = \frac{7}{25} \]

Thus, the value of cos F is \(\frac{7}{25}\).