If 3 cos A = 4 sin A, find the value of :
`3 - cot^2A+ "cosec"^2A`

Answer

To solve the problem, we need to find the value of \( 3 - \cot^2 A + \csc^2 A \) given that \( 3 \cos A = 4 \sin A \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ 3 \cos A = 4 \sin A \]

2. **Rearrange the equation**:
\[ \frac{\cos A}{\sin A} = \frac{4}{3} \]
This implies:
\[ \cot A = \frac{4}{3} \]

3. **Use the identity for cosec and cot**:
We know the identity:
\[ \csc^2 A - \cot^2 A = 1 \]

4. **Express \(\csc^2 A\) in terms of \(\cot^2 A\)**:
Rearranging the identity gives:
\[ \csc^2 A = \cot^2 A + 1 \]

5. **Substitute \(\cot^2 A\) into the expression**:
Since \(\cot A = \frac{4}{3}\), we find \(\cot^2 A\):
\[ \cot^2 A = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \]

6. **Calculate \(\csc^2 A\)**:
\[ \csc^2 A = \cot^2 A + 1 = \frac{16}{9} + 1 = \frac{16}{9} + \frac{9}{9} = \frac{25}{9} \]

7. **Substitute \(\cot^2 A\) and \(\csc^2 A\) into the original expression**:
\[ 3 - \cot^2 A + \csc^2 A = 3 - \frac{16}{9} + \frac{25}{9} \]

8. **Combine the terms**:
\[ 3 = \frac{27}{9} \quad \text{(to have a common denominator)} \]
Therefore,
\[ 3 - \frac{16}{9} + \frac{25}{9} = \frac{27}{9} - \frac{16}{9} + \frac{25}{9} = \frac{27 - 16 + 25}{9} = \frac{36}{9} = 4 \]

9. **Final Result**:
\[ 3 - \cot^2 A + \csc^2 A = 4 \]

### Final Answer:
\[ \boxed{4} \]