If `tan theta =(4)/(3)`, then the value of `sqrt((1+cos theta)/(1-cos theta))` is :

Options

A. 1
B. 2
C. 3
D. 4

Answer

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

### Step-by-Step Solution:

1. **Use the relationship between tangent and secant**:
We know that:
\[ \tan^2 \theta + 1 = \sec^2 \theta \]
Given \( \tan \theta = \frac{4}{3} \), we can calculate \( \sec^2 \theta \):
\[ \tan^2 \theta = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \]
Therefore,
\[ \sec^2 \theta = \tan^2 \theta + 1 = \frac{16}{9} + 1 = \frac{16}{9} + \frac{9}{9} = \frac{25}{9} \]

2. **Find secant**:
Taking the square root gives us:
\[ \sec \theta = \sqrt{\frac{25}{9}} = \frac{5}{3} \]

3. **Relate secant to cosine**:
We know that:
\[ \sec \theta = \frac{1}{\cos \theta} \]
Therefore,
\[ \cos \theta = \frac{1}{\sec \theta} = \frac{3}{5} \]

4. **Substitute cosine into the expression**:
Now we substitute \( \cos \theta \) into the expression \( \sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}} \):
\[ \sqrt{\frac{1 + \frac{3}{5}}{1 - \frac{3}{5}} = \sqrt{\frac{\frac{5}{5} + \frac{3}{5}}{\frac{5}{5} - \frac{3}{5}}} \]
This simplifies to:
\[ \sqrt{\frac{\frac{8}{5}}{\frac{2}{5}}} = \sqrt{\frac{8}{2}} = \sqrt{4} = 2 \]

5. **Final answer**:
Therefore, the value of \( \sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}} \) is \( 2 \).