Number of solutions of the equation `4(cos^(2) 2x+ cos 2 x +1)+tan x (tan x-2sqrt(3))=0` in `[0, 2pi]` is

Options

A. 0
B. 1
C. 2
D. 3

Explanation

To solve the equation \(4(\cos^2 2x + \cos 2x + 1) + \tan x (\tan x - 2\sqrt{3}) = 0\) in the interval \([0, 2\pi]\), we will follow these steps:

### Step 1: Rewrite the equation
The given equation can be rewritten as:
\[ 4\cos^2 2x + 4\cos 2x + 4 + \tan^2 x - 2\sqrt{3}\tan x = 0 \]

### Step 2: Group terms
Rearranging the equation gives us:
\[ 4\cos^2 2x + 4\cos 2x + \tan^2 x - 2\sqrt{3}\tan x + 4 = 0 \]

### Step 3: Identify the conditions for zero
We can express the equation in terms of squares:
\[ (2\cos 2x + 1)^2 + (\tan x - \sqrt{3})^2 = 0 \]
Since both terms are squares, they must be equal to zero:
1. \(2\cos 2x + 1 = 0\)
2. \(\tan x - \sqrt{3} = 0\)

### Step 4: Solve for \(x\) from \(2\cos 2x + 1 = 0\)
From the first equation:
\[ 2\cos 2x = -1 \implies \cos 2x = -\frac{1}{2} \]
The solutions for \(\cos 2x = -\frac{1}{2}\) in the interval \([0, 2\pi]\) are:
\[ 2x = \frac{2\pi}{3}, \frac{4\pi}{3} \implies x = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3} \]

### Step 5: Solve for \(x\) from \(\tan x - \sqrt{3} = 0\)
From the second equation:
\[ \tan x = \sqrt{3} \]
The solutions for \(\tan x = \sqrt{3}\) in the interval \([0, 2\pi]\) are:
\[ x = \frac{\pi}{3}, \frac{4\pi}{3} \]

### Step 6: Find common solutions
Now we find the common solutions from both equations:
- From \(2\cos 2x + 1 = 0\): \(x = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}\)
- From \(\tan x - \sqrt{3} = 0\): \(x = \frac{\pi}{3}, \frac{4\pi}{3}\)

The common solutions are:
1. \(x = \frac{\pi}{3}\)
2. \(x = \frac{4\pi}{3}\)

### Step 7: Count the number of solutions
Thus, the total number of solutions in the interval \([0, 2\pi]\) is \(2\).

### Final Answer:
The number of solutions of the equation in the interval \([0, 2\pi]\) is **2**.

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