`DeltaXYZ` is a right angled at Y. If cotX = 5/12, then what is the value of secZ ?

Options

A. `5//12`
B. `13//5`
C. `13//12`
D. `12//5`

Answer

To solve the problem, we will follow these steps:

1. **Understand the triangle and given information**: We have a right triangle \( \Delta XYZ \) with a right angle at \( Y \). We know that \( \cot X = \frac{5}{12} \).

2. **Identify the sides of the triangle**: The cotangent of an angle is defined as the ratio of the adjacent side to the opposite side. Therefore, if \( \cot X = \frac{5}{12} \), we can assign:
- Adjacent side (base) to angle \( X \) as \( XY = 5 \)
- Opposite side (perpendicular) to angle \( X \) as \( YZ = 12 \)

3. **Calculate the hypotenuse using the Pythagorean theorem**:
\[ XZ = \sqrt{XY^2 + YZ^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]
So, the hypotenuse \( XZ = 13 \).

4. **Find secant of angle \( Z \)**: The secant of an angle is defined as the ratio of the hypotenuse to the adjacent side. For angle \( Z \):
- The hypotenuse \( XZ = 13 \)
- The adjacent side to angle \( Z \) is \( YZ = 12 \)

Therefore,
\[ \sec Z = \frac{XZ}{YZ} = \frac{13}{12} \]

5. **Conclusion**: The value of \( \sec Z \) is \( \frac{13}{12} \).

### Final Answer:
\[ \sec Z = \frac{13}{12} \]