If `0 lt x lt pi` and `cos x + sin x = 1/2`, then tan x is

Options

A. `((4-sqrt(7)))/(3)`
B. `-((4+sqrt(7)))/(3)`
C. `((1+sqrt(7)))/(4)`
D. `((1-sqrt(7)))/(4)`

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:
\[ \cos^2 x + 2\sin x \cos x + \sin^2 x = \frac{1}{4} \]

3. **Use the Pythagorean identity** \(\cos^2 x + \sin^2 x = 1\):
\[ 1 + 2\sin x \cos x = \frac{1}{4} \]

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

5. **Use the double angle identity** for sine:
\[ \sin 2x = 2\sin x \cos x \]
Thus:
\[ \sin 2x = -\frac{3}{4} \]

6. **Set up the equation for \(\tan x\)**:
We know that:
\[ \sin 2x = \frac{2\tan x}{1 + \tan^2 x} \]
Therefore:
\[ \frac{2\tan x}{1 + \tan^2 x} = -\frac{3}{4} \]

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

8. **Rearranging the equation**:
\[ \frac{3}{4}\tan^2 x + 2\tan x + \frac{3}{4} = 0 \]
Multiplying through by 4 to eliminate the fraction:
\[ 3\tan^2 x + 8\tan x + 3 = 0 \]

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

10. **Final values of \(\tan x\)**:
This gives us two possible values:
\[ \tan x = \frac{-4 + \sqrt{7}}{3} \quad \text{and} \quad \tan x = \frac{-4 - \sqrt{7}}{3} \]

Since \(0 < x < \pi\), we need to check which of these values is valid. The first value is valid as it is positive, while the second value is negative.

### Conclusion:
Thus, the value of \(\tan x\) is:
\[ \tan x = \frac{-4 + \sqrt{7}}{3} \]