If `5 sin theta= 3`, the numerical value of `(sec theta - tan theta)/(sec theta + tan theta)` is

Options

A. `1/2`
B. `1/5`
C. `1/3`
D. `1/4`

Answer

To solve the problem, we need to find the value of \((\sec \theta - \tan \theta) / (\sec \theta + \tan \theta)\) given that \(5 \sin \theta = 3\).

### Step-by-Step Solution:

1. **Find \(\sin \theta\)**:
\[ 5 \sin \theta = 3 \implies \sin \theta = \frac{3}{5} \]

**Hint**: Remember that \(\sin \theta\) is the ratio of the opposite side to the hypotenuse in a right triangle.

2. **Use the Pythagorean identity to find \(\cos \theta\)**:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
\[ \left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1 \implies \frac{9}{25} + \cos^2 \theta = 1 \]
\[ \cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25} \implies \cos \theta = \frac{4}{5} \]

**Hint**: The Pythagorean identity is crucial for finding \(\cos \theta\) when you know \(\sin \theta\).

3. **Find \(\sec \theta\) and \(\tan \theta\)**:
\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{4}{5}} = \frac{5}{4} \]
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \]

**Hint**: Remember that \(\sec \theta\) is the reciprocal of \(\cos \theta\) and \(\tan \theta\) is the ratio of \(\sin \theta\) to \(\cos \theta\).

4. **Substitute \(\sec \theta\) and \(\tan \theta\) into the expression**:
\[ \frac{\sec \theta - \tan \theta}{\sec \theta + \tan \theta} = \frac{\frac{5}{4} - \frac{3}{4}}{\frac{5}{4} + \frac{3}{4}} = \frac{\frac{2}{4}}{\frac{8}{4}} = \frac{2}{8} = \frac{1}{4} \]

**Hint**: When combining fractions, ensure you have a common denominator to simplify correctly.

5. **Final Answer**:
The numerical value of \(\frac{\sec \theta - \tan \theta}{\sec \theta + \tan \theta}\) is \(\frac{1}{4}\).

### Summary:
The answer to the problem is \(\frac{1}{4}\).