If `sin theta = 3/5`, then the value of `(tan theta + cos theta)/(cot theta + cosec theta)` 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 1: Determine the values of \(\cos \theta\) and \(\tan \theta\)

Given \(\sin \theta = \frac{3}{5}\), we can use the Pythagorean identity to find \(\cos \theta\).

Using the identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting the value of \(\sin \theta\):
\[ \left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1 \]
\[ \frac{9}{25} + \cos^2 \theta = 1 \]
\[ \cos^2 \theta = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \]
Taking the square root:
\[ \cos \theta = \frac{4}{5} \quad (\text{since } \theta \text{ is in the first quadrant}) \]

Now, we can find \(\tan \theta\):
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \]

### Step 2: Determine the values of \(\cot \theta\) and \(\csc \theta\)

Now, we can find \(\cot \theta\) and \(\csc \theta\):
\[ \cot \theta = \frac{1}{\tan \theta} = \frac{4}{3} \]
\[ \csc \theta = \frac{1}{\sin \theta} = \frac{5}{3} \]

### Step 3: Substitute into the expression

Now we substitute these values into the expression:
\[ \frac{\tan \theta + \cos \theta}{\cot \theta + \csc \theta} = \frac{\frac{3}{4} + \frac{4}{5}}{\frac{4}{3} + \frac{5}{3}} \]

### Step 4: Simplify the numerator and denominator

**Numerator:**
\[ \frac{3}{4} + \frac{4}{5} = \frac{3 \cdot 5 + 4 \cdot 4}{4 \cdot 5} = \frac{15 + 16}{20} = \frac{31}{20} \]

**Denominator:**
\[ \frac{4}{3} + \frac{5}{3} = \frac{4 + 5}{3} = \frac{9}{3} = 3 \]

### Step 5: Final calculation

Now we can substitute back into the expression:
\[ \frac{\frac{31}{20}}{3} = \frac{31}{20} \cdot \frac{1}{3} = \frac{31}{60} \]

Thus, the final answer is:
\[ \frac{31}{60} \]

### Conclusion

The value of \((\tan \theta + \cos \theta) / (\cot \theta + \csc \theta)\) is \(\frac{31}{60}\).

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