The number of solutions of the equation `cotxcosx-1=cotx-cosx, AA in [0, 2pi]` is equal to

Answer

To solve the equation \( \cot x \cos x - 1 = \cot x - \cos x \) for \( x \) in the interval \( [0, 2\pi] \), we can follow these steps:

### Step 1: Rewrite the Equation
Start by rewriting the equation:
\[ \cot x \cos x - 1 = \cot x - \cos x \]
This can be rearranged to:
\[ \cot x \cos x - \cot x + \cos x - 1 = 0 \]

### Step 2: Factor the Equation
We can factor out \( \cot x \) and rearrange the terms:
\[ \cot x (\cos x - 1) + (\cos x - 1) = 0 \]
This gives us:
\[ (\cot x + 1)(\cos x - 1) = 0 \]

### Step 3: Set Each Factor to Zero
Now, we can set each factor to zero:
1. \( \cot x + 1 = 0 \)
2. \( \cos x - 1 = 0 \)

### Step 4: Solve \( \cot x + 1 = 0 \)
From \( \cot x + 1 = 0 \), we have:
\[ \cot x = -1 \]
This implies:
\[ \tan x = -1 \]
The solutions for \( \tan x = -1 \) in the interval \( [0, 2\pi] \) are:
\[ x = \frac{3\pi}{4}, \frac{7\pi}{4} \]

### Step 5: Solve \( \cos x - 1 = 0 \)
From \( \cos x - 1 = 0 \), we have:
\[ \cos x = 1 \]
The solution for \( \cos x = 1 \) in the interval \( [0, 2\pi] \) is:
\[ x = 0, 2\pi \]

### Step 6: Check for Validity of Solutions
Next, we need to check if these solutions are valid in the original equation. The values \( x = 0 \) and \( x = 2\pi \) will make \( \cot x \) undefined because \( \cot 0 \) and \( \cot 2\pi \) both involve division by zero. Therefore, we discard these solutions.

### Step 7: Count the Valid Solutions
The valid solutions we have are:
- From \( \tan x = -1 \): \( x = \frac{3\pi}{4}, \frac{7\pi}{4} \)

Thus, the total number of valid solutions in the interval \( [0, 2\pi] \) is **2**.

### Final Answer
The number of solutions of the equation \( \cot x \cos x - 1 = \cot x - \cos x \) in the interval \( [0, 2\pi] \) is **2**.

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