`DeltaXYZ` is right angled at Y .If cotX`=(3)/(4)` ,what is the value of tan Z?

Options

A. `(5)/(3)`
B. `(4)/(5)`
C. `(4)/(3)`
D. `(3)/(4)`

Answer

To solve the problem, we need to find the value of \( \tan Z \) in triangle \( \Delta XYZ \) which is right-angled at \( Y \) and given that \( \cot X = \frac{3}{4} \).

### Step-by-Step Solution:

1. **Understand the Triangle Configuration**:
- In triangle \( XYZ \), angle \( Y \) is the right angle (90 degrees).
- Therefore, angles \( X \) and \( Z \) are complementary, meaning \( X + Z = 90^\circ \).

2. **Use the Cotangent Definition**:
- We know that \( \cot X = \frac{3}{4} \).
- The cotangent of an angle is defined as the ratio of the adjacent side to the opposite side. Thus, we can express this as:
\[ \cot X = \frac{\text{Adjacent (XY)}}{\text{Opposite (YZ)}} = \frac{3}{4} \]

3. **Assign Values to the Sides**:
- From the cotangent ratio, we can assign:
- \( XY = 3k \) (adjacent side)
- \( YZ = 4k \) (opposite side)
- Here, \( k \) is a common multiplier.

4. **Find the Value of \( \tan Z \)**:
- The tangent of angle \( Z \) is defined as:
\[ \tan Z = \frac{\text{Opposite (XY)}}{\text{Adjacent (YZ)}} \]
- Substituting the values we assigned:
\[ \tan Z = \frac{XY}{YZ} = \frac{3k}{4k} = \frac{3}{4} \]

5. **Conclusion**:
- Therefore, the value of \( \tan Z \) is:
\[ \tan Z = \frac{3}{4} \]