`Delta`PQR is right angled at Q. If cosecP = `25//24`, then what is the value of cotR ?

Options

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

Explanation

To solve the problem, we will use the properties of a right-angled triangle and the definitions of trigonometric ratios.

### Step-by-Step Solution:

1. **Understanding the Triangle**:
- We have a right-angled triangle PQR with the right angle at Q.
- We are given that cosec P = 25/24.

2. **Using the Definition of Cosecant**:
- Cosecant (cosec) is defined as the ratio of the hypotenuse to the opposite side. Therefore:
\[ \text{cosec } P = \frac{\text{Hypotenuse}}{\text{Opposite to } P} \]
- Let the hypotenuse (PQ) = 25k and the opposite side to angle P (QR) = 24k, where k is a scaling factor.

3. **Finding the Third Side**:
- To find the third side (PR), we can use the Pythagorean theorem:
\[ PQ^2 = QR^2 + PR^2 \]
- Substituting the known values:
\[ (25k)^2 = (24k)^2 + PR^2 \]
\[ 625k^2 = 576k^2 + PR^2 \]
\[ PR^2 = 625k^2 - 576k^2 = 49k^2 \]
\[ PR = 7k \]

4. **Identifying the Sides**:
- Now we have:
- Hypotenuse PQ = 25k
- Opposite QR = 24k
- Adjacent PR = 7k

5. **Finding cotangent of angle R**:
- Cotangent (cot) is defined as the ratio of the adjacent side to the opposite side. Therefore:
\[ \text{cot } R = \frac{\text{Adjacent to } R}{\text{Opposite to } R} \]
- For angle R:
- Adjacent side = QR = 24k
- Opposite side = PR = 7k
- Thus:
\[ \text{cot } R = \frac{QR}{PR} = \frac{24k}{7k} = \frac{24}{7} \]

6. **Final Answer**:
- Therefore, the value of cot R is:
\[ \text{cot } R = \frac{24}{7} \]