Solve `cos x gt 1 -(2x)/pi`.

Explanation

To solve the inequality \( \cos x > 1 - \frac{2x}{\pi} \), we will follow these steps:

### Step 1: Rewrite the inequality
We start with the inequality:
\[ \cos x > 1 - \frac{2x}{\pi} \]

### Step 2: Set up the equation
To find the points where the two sides are equal, we set:
\[ \cos x = 1 - \frac{2x}{\pi} \]

### Step 3: Find the critical points
We can rearrange this equation:
\[ \cos x + \frac{2x}{\pi} - 1 = 0 \]
Now, we will find the points where \( \cos x \) intersects the line \( y = 1 - \frac{2x}{\pi} \).

### Step 4: Analyze the function
The function \( \cos x \) oscillates between -1 and 1, while the line \( y = 1 - \frac{2x}{\pi} \) is a straight line with a negative slope. We need to find the intersection points.

### Step 5: Find specific values
1. When \( x = 0 \):
\[ \cos(0) = 1 \quad \text{and} \quad 1 - \frac{2(0)}{\pi} = 1 \]
So, \( \cos(0) = 1 \).

2. When \( x = \frac{\pi}{2} \):
\[ \cos\left(\frac{\pi}{2}\right) = 0 \quad \text{and} \quad 1 - \frac{2\left(\frac{\pi}{2}\right)}{\pi} = 0 \]
So, \( \cos\left(\frac{\pi}{2}\right) = 0 \).

3. When \( x = \pi \):
\[ \cos(\pi) = -1 \quad \text{and} \quad 1 - \frac{2\pi}{\pi} = -1 \]
So, \( \cos(\pi) = -1 \).

### Step 6: Determine intervals
Now, we will check the intervals between these points:
- From \( 0 \) to \( \frac{\pi}{2} \): \( \cos x \) decreases from \( 1 \) to \( 0 \).
- From \( \frac{\pi}{2} \) to \( \pi \): \( \cos x \) decreases from \( 0 \) to \( -1 \).

### Step 7: Analyze the inequality
We need to find where \( \cos x > 1 - \frac{2x}{\pi} \):
- In the interval \( [0, \frac{\pi}{2}] \), \( \cos x \) is greater than \( 1 - \frac{2x}{\pi} \).
- In the interval \( [\frac{\pi}{2}, \pi] \), \( \cos x \) is equal to \( 1 - \frac{2x}{\pi} \) at \( x = \frac{\pi}{2} \) and less than it afterwards.

### Step 8: Conclusion
The solution to the inequality \( \cos x > 1 - \frac{2x}{\pi} \) is:
\[ x \in [0, \frac{\pi}{2}) \cup (\pi, \infty) \]