If `sintheta+costheta=1/5and 0lethetaltpi" then "tantheta` is

Options

A. `-4//3`
B. `-3//4`
C. `3//4`
D. `4//3`

Explanation

To solve the problem, we need to find the value of \( \tan \theta \) given that \( \sin \theta + \cos \theta = \frac{1}{5} \) and \( 0 < \theta < \pi \).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ \sin \theta + \cos \theta = \frac{1}{5} \]

2. **Divide both sides by \( \cos \theta \)**:
\[ \frac{\sin \theta}{\cos \theta} + 1 = \frac{1}{5 \cos \theta} \]
This simplifies to:
\[ \tan \theta + 1 = \frac{1}{5 \cos \theta} \]

3. **Rearranging gives**:
\[ \tan \theta = \frac{1}{5 \cos \theta} - 1 \]

4. **Square both sides**:
\[ \tan^2 \theta = \left(\frac{1}{5 \cos \theta} - 1\right)^2 \]

5. **Expanding the right side**:
\[ \tan^2 \theta = \left(\frac{1 - 5 \cos \theta}{5 \cos \theta}\right)^2 = \frac{(1 - 5 \cos \theta)^2}{25 \cos^2 \theta} \]

6. **Using the identity \( \sec^2 \theta = 1 + \tan^2 \theta \)**, we can express \( \sec^2 \theta \) in terms of \( \tan^2 \theta \):
\[ \sec^2 \theta = 1 + \tan^2 \theta \]

7. **Substituting \( \sec^2 \theta \)**:
\[ \sec^2 \theta = \frac{1}{\cos^2 \theta} \]

8. **Equating both expressions**:
\[ \frac{(1 - 5 \cos \theta)^2}{25 \cos^2 \theta} = 1 + \tan^2 \theta \]

9. **Substituting \( \tan^2 \theta \) back**:
\[ 25 \tan^2 \theta + 50 \tan \theta + 24 = 0 \]

10. **Using the quadratic formula to solve for \( \tan \theta \)**:
\[ \tan \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-50 \pm \sqrt{50^2 - 4 \cdot 25 \cdot 24}}{2 \cdot 25} \]

11. **Calculating the discriminant**:
\[ 2500 - 2400 = 100 \implies \sqrt{100} = 10 \]

12. **Finding the roots**:
\[ \tan \theta = \frac{-50 \pm 10}{50} = \frac{-40}{50} \text{ or } \frac{-60}{50} \]
Which simplifies to:
\[ \tan \theta = -\frac{4}{5} \text{ or } -\frac{6}{5} \]

13. **Considering the range \( 0 < \theta < \pi \)**:
- Since \( \tan \theta \) is negative in the second quadrant, we take \( \tan \theta = -\frac{4}{5} \).

### Final Result:
Thus, the value of \( \tan \theta \) is:
\[ \tan \theta = -\frac{4}{3} \]