If `5 sin theta = 3,` the numerical value of`(sec theta - tan theta)/(sec theta + tan theta)` is
यदि `5 sin theta= 3,` समीकरण का संख्यात्मक मान है `(sec theta - tan theta)/(sec theta + tan theta)`

Options

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

Explanation

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

### Step-by-step Solution:

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

2. **Construct a right triangle**:
- In a right triangle, let the opposite side (perpendicular) be \(3k\) and the hypotenuse be \(5k\). Here, \(k\) is a scaling factor.
- To find the adjacent side (base), we use the Pythagorean theorem:
\[ \text{Base} = \sqrt{(\text{Hypotenuse})^2 - (\text{Opposite})^2} = \sqrt{(5k)^2 - (3k)^2} = \sqrt{25k^2 - 9k^2} = \sqrt{16k^2} = 4k \]

3. **Calculate \(\sec \theta\) and \(\tan \theta\)**:
- \(\sec \theta = \frac{\text{Hypotenuse}}{\text{Base}} = \frac{5k}{4k} = \frac{5}{4}\)
- \(\tan \theta = \frac{\text{Opposite}}{\text{Base}} = \frac{3k}{4k} = \frac{3}{4}\)

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

5. **Simplify the expression**:
- The numerator:
\[ \frac{5}{4} - \frac{3}{4} = \frac{2}{4} = \frac{1}{2} \]
- The denominator:
\[ \frac{5}{4} + \frac{3}{4} = \frac{8}{4} = 2 \]

6. **Final calculation**:
\[ \frac{\frac{1}{2}}{2} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \]

Thus, the numerical value of \((\sec \theta - \tan \theta)/(\sec \theta + \tan \theta)\) is \(\frac{1}{4}\).