If `log_(0.5) sin x=1-log_(0.5) cos x,` then the number of solutions of `x in[-2pi,2pi]` is

Explanation

To solve the equation \( \log_{0.5} \sin x = 1 - \log_{0.5} \cos x \), we will follow these steps:

### Step 1: Rewrite the logarithmic equation
We start with the equation:
\[ \log_{0.5} \sin x = 1 - \log_{0.5} \cos x \]
Using the property of logarithms \( \log_a b + \log_a c = \log_a (bc) \), we can rewrite the right side:
\[ \log_{0.5} \sin x + \log_{0.5} \cos x = 1 \]
This can be rewritten as:
\[ \log_{0.5} (\sin x \cos x) = 1 \]

### Step 2: Convert the logarithmic equation to exponential form
Using the definition of logarithms, we convert the equation to its exponential form:
\[ \sin x \cos x = 0.5^1 = 0.5 \]

### Step 3: Use the double angle identity
We know that \( \sin x \cos x = \frac{1}{2} \sin(2x) \), so we can rewrite the equation:
\[ \frac{1}{2} \sin(2x) = 0.5 \]
Multiplying both sides by 2 gives:
\[ \sin(2x) = 1 \]

### Step 4: Solve for \( 2x \)
The general solution for \( \sin(2x) = 1 \) is:
\[ 2x = \frac{\pi}{2} + 2k\pi \quad \text{for } k \in \mathbb{Z} \]
Dividing by 2, we find:
\[ x = \frac{\pi}{4} + k\pi \]

### Step 5: Determine the values of \( x \) in the interval \([-2\pi, 2\pi]\)
Now we will find the values of \( k \) such that \( x \) lies within the interval \([-2\pi, 2\pi]\).

1. For \( k = -2 \):
\[ x = \frac{\pi}{4} - 2\pi = \frac{\pi}{4} - \frac{8\pi}{4} = -\frac{7\pi}{4} \]
2. For \( k = -1 \):
\[ x = \frac{\pi}{4} - \pi = \frac{\pi}{4} - \frac{4\pi}{4} = -\frac{3\pi}{4} \]
3. For \( k = 0 \):
\[ x = \frac{\pi}{4} \]
4. For \( k = 1 \):
\[ x = \frac{\pi}{4} + \pi = \frac{\pi}{4} + \frac{4\pi}{4} = \frac{5\pi}{4} \]
5. For \( k = 2 \):
\[ x = \frac{\pi}{4} + 2\pi = \frac{\pi}{4} + \frac{8\pi}{4} = \frac{9\pi}{4} \quad (\text{not in } [-2\pi, 2\pi]) \]

### Step 6: List the valid solutions
The valid solutions in the interval \([-2\pi, 2\pi]\) are:
- \( x = -\frac{7\pi}{4} \)
- \( x = -\frac{3\pi}{4} \)
- \( x = \frac{\pi}{4} \)
- \( x = \frac{5\pi}{4} \)

### Conclusion
Thus, the total number of solutions in the interval \([-2\pi, 2\pi]\) is **4**.

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