If `3 tan theta =4`, then find `sqrt( (1- sin theta)/(1 + sin theta))`

Options

A. `1/3`
B. `1/2`
C. `2/3`
D. none to these

Answer

To solve the problem, we need to find the value of \( \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} \) given that \( 3 \tan \theta = 4 \).

### Step-by-Step Solution:

1. **Find \( \tan \theta \)**:
\[ \tan \theta = \frac{4}{3} \]

2. **Identify the sides of the triangle**:
- We can represent \( \tan \theta \) as \( \frac{\text{opposite}}{\text{adjacent}} \). Here, the opposite side (perpendicular) is 4 and the adjacent side (base) is 3.

3. **Use the Pythagorean theorem to find the hypotenuse**:
\[ \text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2 \]
\[ \text{hypotenuse}^2 = 4^2 + 3^2 = 16 + 9 = 25 \]
\[ \text{hypotenuse} = \sqrt{25} = 5 \]

4. **Calculate \( \sin \theta \)**:
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{5} \]

5. **Substitute \( \sin \theta \) into the expression**:
\[ \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} = \sqrt{\frac{1 - \frac{4}{5}}{1 + \frac{4}{5}}} \]

6. **Simplify the expression**:
- Calculate \( 1 - \frac{4}{5} \):
\[ 1 - \frac{4}{5} = \frac{1}{5} \]
- Calculate \( 1 + \frac{4}{5} \):
\[ 1 + \frac{4}{5} = \frac{9}{5} \]
- Now substitute back:
\[ \sqrt{\frac{\frac{1}{5}}{\frac{9}{5}}} = \sqrt{\frac{1}{5} \cdot \frac{5}{9}} = \sqrt{\frac{1}{9}} = \frac{1}{3} \]

### Final Answer:
\[ \sqrt{\frac{1 - \sin \theta}{1 + \sin \theta}} = \frac{1}{3} \]