Let `f(x)=sin^(23)x-cos^(22)xa n dg(x)=1+1/2tan^(-1)|x|` . Then the number of values of `x` in the interval `[-10pi,8pi]` satisfying the equation `f(x)=sgn(g(x))` is __________

Explanation

To solve the problem, we need to find the number of values of \( x \) in the interval \([-10\pi, 8\pi]\) that satisfy the equation \( f(x) = \text{sgn}(g(x)) \), where \( f(x) = \sin^{23}(x) - \cos^{22}(x) \) and \( g(x) = 1 + \frac{1}{2} \tan^{-1} |x| \).

### Step-by-Step Solution:

1. **Understanding the Signum Function:**
The signum function, \( \text{sgn}(x) \), is defined as:
- \( \text{sgn}(x) = -1 \) if \( x < 0 \)
- \( \text{sgn}(x) = 0 \) if \( x = 0 \)
- \( \text{sgn}(x) = 1 \) if \( x > 0 \)

2. **Analyzing \( g(x) \):**
The function \( g(x) = 1 + \frac{1}{2} \tan^{-1} |x| \):
- Since \( \tan^{-1} |x| \) is always non-negative, \( g(x) \) is always greater than or equal to 1.
- Therefore, \( \text{sgn}(g(x)) = 1 \) for all \( x \).

3. **Setting Up the Equation:**
Since \( \text{sgn}(g(x)) = 1 \), the equation simplifies to:
\[ f(x) = 1 \]
This means we need to solve:
\[ \sin^{23}(x) - \cos^{22}(x) = 1 \]

4. **Rearranging the Equation:**
Rearranging gives us:
\[ \sin^{23}(x) = 1 + \cos^{22}(x) \]
Since \( \sin^{23}(x) \) can only equal 1 when \( \sin(x) = 1 \) (which occurs at \( x = \frac{\pi}{2} + 2n\pi \)), we need to check if this satisfies the equation.

5. **Finding the Values of \( x \):**
For \( \sin(x) = 1 \):
\[ x = \frac{\pi}{2} + 2n\pi \quad (n \in \mathbb{Z}) \]

6. **Determining the Range of \( n \):**
We need to find \( n \) such that:
\[ -10\pi \leq \frac{\pi}{2} + 2n\pi \leq 8\pi \]
Dividing through by \( \pi \):
\[ -10 \leq \frac{1}{2} + 2n \leq 8 \]
Rearranging gives:
\[ -10 - \frac{1}{2} \leq 2n \leq 8 - \frac{1}{2} \]
Simplifying:
\[ -\frac{21}{2} \leq 2n \leq \frac{15}{2} \]
Dividing by 2:
\[ -\frac{21}{4} \leq n \leq \frac{15}{4} \]

7. **Finding Integer Values of \( n \):**
The integer values of \( n \) in the range \( -5.25 \leq n \leq 3.75 \) are:
\[ n = -5, -4, -3, -2, -1, 0, 1, 2, 3 \]
This gives us a total of 9 integer values.

### Conclusion:
The number of values of \( x \) in the interval \([-10\pi, 8\pi]\) satisfying the equation \( f(x) = \text{sgn}(g(x)) \) is **9**.