If ` 0 < x < pi` and `cosx + sinx = 1/2` then ` tanx =`

Options

A. `(4-sqrt(7))/(3)`
B. `(4+sqrt(7))/(3)`
C. `(-(4+sqrt(7)))/(3)`
D. `(-4+sqrt(7))/(3)`

Answer

To solve the problem, we need to find the value of \( \tan x \) given that \( \cos x + \sin x = \frac{1}{2} \) and \( 0 < x < \pi \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \cos x + \sin x = \frac{1}{2} \]

2. **Square both sides:**
\[ (\cos x + \sin x)^2 = \left(\frac{1}{2}\right)^2 \]
This expands to:
\[ \cos^2 x + \sin^2 x + 2 \sin x \cos x = \frac{1}{4} \]

3. **Use the Pythagorean identity:**
Since \( \cos^2 x + \sin^2 x = 1 \), we can substitute this into the equation:
\[ 1 + 2 \sin x \cos x = \frac{1}{4} \]

4. **Rearrange the equation:**
\[ 2 \sin x \cos x = \frac{1}{4} - 1 \]
\[ 2 \sin x \cos x = -\frac{3}{4} \]

5. **Recognize that \( \sin 2x = 2 \sin x \cos x \):**
\[ \sin 2x = -\frac{3}{4} \]

6. **Express \( \sin 2x \) in terms of \( \tan x \):**
We know that:
\[ \sin 2x = \frac{2 \tan x}{1 + \tan^2 x} \]
Therefore, we can set up the equation:
\[ \frac{2 \tan x}{1 + \tan^2 x} = -\frac{3}{4} \]

7. **Cross-multiply to eliminate the fraction:**
\[ 2 \tan x = -\frac{3}{4}(1 + \tan^2 x) \]
\[ 2 \tan x = -\frac{3}{4} - \frac{3}{4} \tan^2 x \]

8. **Rearrange the equation:**
Multiply through by 4 to eliminate the fraction:
\[ 8 \tan x = -3 - 3 \tan^2 x \]
Rearranging gives:
\[ 3 \tan^2 x + 8 \tan x + 3 = 0 \]

9. **Use the quadratic formula to solve for \( \tan x \):**
The quadratic formula is given by:
\[ \tan x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 3 \), \( b = 8 \), and \( c = 3 \):
\[ \tan x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 3 \cdot 3}}{2 \cdot 3} \]
\[ \tan x = \frac{-8 \pm \sqrt{64 - 36}}{6} \]
\[ \tan x = \frac{-8 \pm \sqrt{28}}{6} \]
\[ \tan x = \frac{-8 \pm 2\sqrt{7}}{6} \]
Simplifying gives:
\[ \tan x = \frac{-4 \pm \sqrt{7}}{3} \]

10. **Determine the valid solution based on the quadrant:**
Since \( 0 < x < \pi \), we need to check the signs of \( \tan x \):
- In the first quadrant \( (0, \frac{\pi}{2}) \), \( \tan x \) is positive.
- In the second quadrant \( (\frac{\pi}{2}, \pi) \), \( \tan x \) is negative.

Thus, we take the negative solution:
\[ \tan x = \frac{-4 - \sqrt{7}}{3} \]

### Final Answer:
\[ \tan x = \frac{-4 - \sqrt{7}}{3} \]