Given : `tan A=4/3 ` , find : `("cosec"A)/(cot A- secA)`

Answer

To solve the problem, we need to find the value of \(\frac{\csc A}{\cot A - \sec A}\) given that \(\tan A = \frac{4}{3}\).

### Step-by-step Solution:

1. **Understand the given information**:
We know that \(\tan A = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3}\). This means in a right triangle, the opposite side (perpendicular) is 4 and the adjacent side (base) is 3.

2. **Find the hypotenuse**:
Using the Pythagorean theorem:
\[ h = \sqrt{(\text{perpendicular})^2 + (\text{base})^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
So, the hypotenuse \(h = 5\).

3. **Calculate \(\csc A\)**:
\(\csc A\) is defined as:
\[ \csc A = \frac{\text{hypotenuse}}{\text{opposite}} = \frac{5}{4} \]

4. **Calculate \(\cot A\)**:
\(\cot A\) is defined as:
\[ \cot A = \frac{\text{adjacent}}{\text{opposite}} = \frac{3}{4} \]

5. **Calculate \(\sec A\)**:
\(\sec A\) is defined as:
\[ \sec A = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{5}{3} \]

6. **Substitute values into the expression**:
Now we substitute \(\csc A\), \(\cot A\), and \(\sec A\) into the expression:
\[ \frac{\csc A}{\cot A - \sec A} = \frac{\frac{5}{4}}{\frac{3}{4} - \frac{5}{3}} \]

7. **Simplify the denominator**:
To simplify \(\cot A - \sec A\), we need a common denominator:
\[ \frac{3}{4} - \frac{5}{3} = \frac{3 \times 3}{4 \times 3} - \frac{5 \times 4}{3 \times 4} = \frac{9}{12} - \frac{20}{12} = \frac{9 - 20}{12} = \frac{-11}{12} \]

8. **Final substitution**:
Now substitute back into the expression:
\[ \frac{\frac{5}{4}}{\frac{-11}{12}} = \frac{5}{4} \times \frac{-12}{11} = \frac{5 \times -12}{4 \times 11} = \frac{-60}{44} \]

9. **Simplify the fraction**:
Simplifying \(\frac{-60}{44}\):
\[ \frac{-60}{44} = \frac{-15}{11} \]

### Final Answer:
Thus, the final answer is:
\[ \frac{\csc A}{\cot A - \sec A} = \frac{-15}{11} \]