If `sinA=(12)/(13)`, then the value of (sec A+cosec A) is :

Options

A. `(67)/(20)`
B. `(221)/(60)`
C. `(11)/3`
D. `(211)/(60)`

Answer

To solve the problem, we need to find the value of \( \sec A + \csc A \) given that \( \sin A = \frac{12}{13} \).

### Step-by-Step Solution:

1. **Identify the Given Value**:
We know that \( \sin A = \frac{12}{13} \).

2. **Use the Pythagorean Identity**:
We can use the identity \( \sin^2 A + \cos^2 A = 1 \) to find \( \cos A \).
\[ \sin^2 A = \left(\frac{12}{13}\right)^2 = \frac{144}{169} \]
Therefore,
\[ \cos^2 A = 1 - \sin^2 A = 1 - \frac{144}{169} = \frac{169 - 144}{169} = \frac{25}{169} \]

3. **Calculate \( \cos A \)**:
Taking the square root of both sides, we find:
\[ \cos A = \sqrt{\frac{25}{169}} = \frac{5}{13} \]

4. **Find \( \sec A \) and \( \csc A \)**:
- \( \sec A = \frac{1}{\cos A} = \frac{1}{\frac{5}{13}} = \frac{13}{5} \)
- \( \csc A = \frac{1}{\sin A} = \frac{1}{\frac{12}{13}} = \frac{13}{12} \)

5. **Add \( \sec A \) and \( \csc A \)**:
Now we can find \( \sec A + \csc A \):
\[ \sec A + \csc A = \frac{13}{5} + \frac{13}{12} \]

6. **Find a Common Denominator**:
The least common multiple of 5 and 12 is 60. We convert both fractions:
\[ \frac{13}{5} = \frac{13 \times 12}{5 \times 12} = \frac{156}{60} \]
\[ \frac{13}{12} = \frac{13 \times 5}{12 \times 5} = \frac{65}{60} \]

7. **Add the Two Fractions**:
Now we can add the two fractions:
\[ \sec A + \csc A = \frac{156}{60} + \frac{65}{60} = \frac{156 + 65}{60} = \frac{221}{60} \]

### Final Answer:
Thus, the value of \( \sec A + \csc A \) is \( \frac{221}{60} \).