If `sintheta=(3)/(5)` , then the value of `(tantheta+costheta)/(cottheta+cosectheta)` is equal to

Options

A. `(29)/(60)`
B. `(31)/(60)`
C. `(34)/(60)`
D. `(37)/(60)`

Answer

To solve the problem, we need to find the value of \((\tan \theta + \cos \theta) / (\cot \theta + \csc \theta)\) given that \(\sin \theta = \frac{3}{5}\).

### Step-by-Step Solution:

1. **Construct a Right Triangle**:
- Given \(\sin \theta = \frac{3}{5}\), we can visualize this in a right triangle where the opposite side (perpendicular) is 3 and the hypotenuse is 5.

2. **Find the Base**:
- Using the Pythagorean theorem:
\[ \text{Base} = \sqrt{\text{Hypotenuse}^2 - \text{Perpendicular}^2} = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \]

3. **Calculate \(\tan \theta\)**:
- \(\tan \theta = \frac{\text{Perpendicular}}{\text{Base}} = \frac{3}{4}\)

4. **Calculate \(\cos \theta\)**:
- \(\cos \theta = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{4}{5}\)

5. **Calculate \(\cot \theta\)**:
- \(\cot \theta = \frac{1}{\tan \theta} = \frac{4}{3}\)

6. **Calculate \(\csc \theta\)**:
- \(\csc \theta = \frac{1}{\sin \theta} = \frac{5}{3}\)

7. **Substitute Values into the Expression**:
- Now, substitute the values into the expression:
\[ \frac{\tan \theta + \cos \theta}{\cot \theta + \csc \theta} = \frac{\frac{3}{4} + \frac{4}{5}}{\frac{4}{3} + \frac{5}{3}} \]

8. **Simplify the Numerator**:
- To add \(\frac{3}{4}\) and \(\frac{4}{5}\), find a common denominator (20):
\[ \frac{3}{4} = \frac{15}{20}, \quad \frac{4}{5} = \frac{16}{20} \quad \Rightarrow \quad \frac{15}{20} + \frac{16}{20} = \frac{31}{20} \]

9. **Simplify the Denominator**:
- For \(\frac{4}{3} + \frac{5}{3}\):
\[ \frac{4}{3} + \frac{5}{3} = \frac{9}{3} = 3 \]

10. **Final Calculation**:
- Now, substitute back into the expression:
\[ \frac{\frac{31}{20}}{3} = \frac{31}{20} \times \frac{1}{3} = \frac{31}{60} \]

Thus, the value of \(\frac{\tan \theta + \cos \theta}{\cot \theta + \csc \theta}\) is \(\frac{31}{60}\).

### Final Answer:
\[ \frac{31}{60} \]