Given : `cos A=5/13` evaluate :
`cotA+1/(cosA)`

Answer

To solve the problem, we are given that \( \cos A = \frac{5}{13} \) and we need to evaluate \( \cot A + \frac{1}{\cos A} \).

### Step-by-Step Solution:

1. **Identify the Triangle**:
We know that \( \cos A = \frac{\text{Base}}{\text{Hypotenuse}} \). Here, the base is 5 and the hypotenuse is 13. We can represent this in a right triangle where:
- Base (adjacent side) = 5
- Hypotenuse = 13

2. **Find the Perpendicular Side**:
To find the perpendicular side (opposite side), we can use the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Base}^2 + \text{Perpendicular}^2 \]
Rearranging gives:
\[ \text{Perpendicular}^2 = \text{Hypotenuse}^2 - \text{Base}^2 \]
Plugging in the values:
\[ \text{Perpendicular}^2 = 13^2 - 5^2 = 169 - 25 = 144 \]
Taking the square root:
\[ \text{Perpendicular} = \sqrt{144} = 12 \]

3. **Calculate \( \cot A \)**:
The cotangent of angle A is given by:
\[ \cot A = \frac{\text{Base}}{\text{Perpendicular}} = \frac{5}{12} \]

4. **Calculate \( \frac{1}{\cos A} \)**:
We know \( \cos A = \frac{5}{13} \), so:
\[ \frac{1}{\cos A} = \frac{1}{\frac{5}{13}} = \frac{13}{5} \]

5. **Combine the Results**:
Now we can evaluate \( \cot A + \frac{1}{\cos A} \):
\[ \cot A + \frac{1}{\cos A} = \frac{5}{12} + \frac{13}{5} \]

6. **Finding a Common Denominator**:
The least common multiple (LCM) of 12 and 5 is 60. We convert each fraction:
\[ \frac{5}{12} = \frac{5 \times 5}{12 \times 5} = \frac{25}{60} \]
\[ \frac{13}{5} = \frac{13 \times 12}{5 \times 12} = \frac{156}{60} \]

7. **Add the Fractions**:
Now we can add the two fractions:
\[ \frac{25}{60} + \frac{156}{60} = \frac{25 + 156}{60} = \frac{181}{60} \]

### Final Answer:
Thus, the value of \( \cot A + \frac{1}{\cos A} \) is \( \frac{181}{60} \).