`cot^(2) theta - (1 + sqrt3) cot theta + sqrt3 = 0,0 lt theta lt (pi)/(2)`

Answer

To solve the equation \( \cot^2 \theta - (1 + \sqrt{3}) \cot \theta + \sqrt{3} = 0 \) for \( 0 < \theta < \frac{\pi}{2} \), we will follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ \cot^2 \theta - (1 + \sqrt{3}) \cot \theta + \sqrt{3} = 0 \]

### Step 2: Let \( x = \cot \theta \)
We can substitute \( x \) for \( \cot \theta \):
\[ x^2 - (1 + \sqrt{3})x + \sqrt{3} = 0 \]

### Step 3: Use the quadratic formula
To solve for \( x \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = -(1 + \sqrt{3}) \), and \( c = \sqrt{3} \).

### Step 4: Calculate the discriminant
First, calculate the discriminant \( b^2 - 4ac \):
\[ b^2 = (1 + \sqrt{3})^2 = 1 + 2\sqrt{3} + 3 = 4 + 2\sqrt{3} \]
\[ 4ac = 4 \cdot 1 \cdot \sqrt{3} = 4\sqrt{3} \]
Now, the discriminant is:
\[ D = (4 + 2\sqrt{3}) - 4\sqrt{3} = 4 - 2\sqrt{3} \]

### Step 5: Solve for \( x \)
Now plug the values into the quadratic formula:
\[ x = \frac{(1 + \sqrt{3}) \pm \sqrt{4 - 2\sqrt{3}}}{2} \]

### Step 6: Simplify the square root
To simplify \( \sqrt{4 - 2\sqrt{3}} \), we can express it as:
\[ \sqrt{4 - 2\sqrt{3}} = \sqrt{(\sqrt{3} - 1)^2} = \sqrt{3} - 1 \]

### Step 7: Substitute back into the formula
Now substituting back, we have:
\[ x = \frac{(1 + \sqrt{3}) \pm (\sqrt{3} - 1)}{2} \]

### Step 8: Calculate the two possible values for \( x \)
1. For the positive case:
\[ x_1 = \frac{(1 + \sqrt{3}) + (\sqrt{3} - 1)}{2} = \frac{2\sqrt{3}}{2} = \sqrt{3} \]
2. For the negative case:
\[ x_2 = \frac{(1 + \sqrt{3}) - (\sqrt{3} - 1)}{2} = \frac{2}{2} = 1 \]

### Step 9: Find \( \theta \)
Now we have two values for \( x \):
1. \( x = \sqrt{3} \) implies \( \cot \theta = \sqrt{3} \) which gives \( \theta = \frac{\pi}{6} \).
2. \( x = 1 \) implies \( \cot \theta = 1 \) which gives \( \theta = \frac{\pi}{4} \).

### Step 10: Conclusion
Thus, the solutions for \( \theta \) in the interval \( 0 < \theta < \frac{\pi}{2} \) are:
\[ \theta = \frac{\pi}{6}, \frac{\pi}{4} \]