`DeltaXYZ` is right angled at Y. If cosec `X=(17)/(15)` ,then what is the value of cotZ?

Options

A. `(17)/(15)`
B. `(8)/(17)`
C. `(17)/(8)`
D. `(15)/(8)`

Answer

To solve the problem, we need to find the value of cot Z in the right-angled triangle XYZ, where the right angle is at Y and cosec X = 17/15.

### Step-by-Step Solution:

1. **Understanding the Triangle**:
- We have triangle XYZ, with the right angle at Y.
- Let angle X be represented as θ and angle Z as 90° - θ.

2. **Using Cosecant**:
- We know that cosec X = 1/sin X.
- Given that cosec X = 17/15, we can find sin X:
\[ \sin X = \frac{1}{\text{cosec } X} = \frac{15}{17} \]

3. **Identifying Sides**:
- In a right triangle, we can relate the sides to the angles.
- Let the hypotenuse (XZ) be 17 and the opposite side to angle X (YZ) be 15.
- We can find the adjacent side (XY) using the Pythagorean theorem:
\[ XZ^2 = XY^2 + YZ^2 \]
\[ 17^2 = XY^2 + 15^2 \]

4. **Calculating XY**:
- Substitute the values:
\[ 289 = XY^2 + 225 \]
\[ XY^2 = 289 - 225 = 64 \]
\[ XY = \sqrt{64} = 8 \]

5. **Finding Cotangent of Z**:
- Cotangent of angle Z is defined as:
\[ \cot Z = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{XY}{YZ} \]
- Substitute the values:
\[ \cot Z = \frac{8}{15} \]

6. **Final Answer**:
- The value of cot Z is:
\[ \cot Z = \frac{15}{8} \]

### Summary:
Thus, the value of cot Z is \( \frac{15}{8} \).