If `(cot^(2)x)/((1+sqrt(3)))+(1)/(2)(3-sqrt(3))=cotx`, then what is the value of x ?

Options

A. `(pi)/(2),(pi)/(4)`
B. `(pi)/(6),(pi)/(2)`
C. `(pi)/(6),(pi)/(4)`
D. `pi,(pi)/(4)`

Answer

To solve the equation

\[ \frac{\cot^2 x}{1 + \sqrt{3}} + \frac{1}{2(3 - \sqrt{3})} = \cot x, \]

we will follow these steps:

### Step 1: Simplify the equation
First, we will find a common denominator for the left-hand side. The common denominator is \((1 + \sqrt{3})(2(3 - \sqrt{3}))\).

So, we rewrite the equation as:

\[ \frac{\cot^2 x \cdot 2(3 - \sqrt{3}) + (1 + \sqrt{3})}{(1 + \sqrt{3})(2(3 - \sqrt{3}))} = \cot x. \]

### Step 2: Clear the denominator
Multiply both sides by the common denominator:

\[ \cot^2 x \cdot 2(3 - \sqrt{3}) + (1 + \sqrt{3}) = \cot x \cdot (1 + \sqrt{3})(2(3 - \sqrt{3})). \]

### Step 3: Expand both sides
Now, we will expand both sides of the equation.

Left-hand side:

\[ \cot^2 x \cdot 2(3 - \sqrt{3}) + (1 + \sqrt{3}). \]

Right-hand side:

\[ \cot x \cdot (1 + \sqrt{3})(2(3 - \sqrt{3})) = \cot x \cdot (2(3 - \sqrt{3}) + 2\sqrt{3}(3 - \sqrt{3})). \]

### Step 4: Rearranging the equation
Rearranging gives us a quadratic equation in terms of \(\cot x\):

\[ \cot^2 x \cdot 2(3 - \sqrt{3}) - \cot x \cdot (1 + \sqrt{3})(2(3 - \sqrt{3})) + (1 + \sqrt{3}) = 0. \]

### Step 5: Solve the quadratic equation
Let \(a = 2(3 - \sqrt{3})\), \(b = -(1 + \sqrt{3})(2(3 - \sqrt{3}))\), and \(c = (1 + \sqrt{3})\). We will use the quadratic formula:

\[ \cot x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]

### Step 6: Find the values of \(x\)
Once we find \(\cot x\), we can determine the angles \(x\) using the identity:

\[ x = \cot^{-1}(\cot x). \]

### Step 7: Check possible values
The possible values of \(x\) based on the options given are \(\frac{\pi}{2}, \frac{\pi}{4}, \frac{\pi}{6}, \frac{\pi}{8}, \pi\).