Solve the inequality `sin2x gtsqrt(2)sin^2x+(2-sqrt(2))cos^2x`

Explanation

To solve the inequality \( \sin 2x > \sqrt{2} \sin^2 x + (2 - \sqrt{2}) \cos^2 x \), we will follow these steps:

### Step 1: Rewrite the inequality
We start with the given inequality:
\[ \sin 2x > \sqrt{2} \sin^2 x + (2 - \sqrt{2}) \cos^2 x \]

### Step 2: Use the identity for \(\sin 2x\)
Recall that \( \sin 2x = 2 \sin x \cos x \). We can substitute this into the inequality:
\[ 2 \sin x \cos x > \sqrt{2} \sin^2 x + (2 - \sqrt{2}) \cos^2 x \]

### Step 3: Express \(\cos^2 x\) in terms of \(\sin^2 x\)
Using the identity \( \cos^2 x = 1 - \sin^2 x \), we can rewrite the right side:
\[ 2 \sin x \cos x > \sqrt{2} \sin^2 x + (2 - \sqrt{2})(1 - \sin^2 x) \]
This simplifies to:
\[ 2 \sin x \cos x > \sqrt{2} \sin^2 x + (2 - \sqrt{2}) - (2 - \sqrt{2}) \sin^2 x \]

### Step 4: Combine like terms
Combine the terms involving \(\sin^2 x\):
\[ 2 \sin x \cos x > (2 - \sqrt{2} + \sqrt{2}) \sin^2 x + (2 - \sqrt{2}) \]
This simplifies to:
\[ 2 \sin x \cos x > 2 \sin^2 x + (2 - \sqrt{2}) \]

### Step 5: Rearrange the inequality
Rearranging gives:
\[ 2 \sin x \cos x - 2 \sin^2 x - (2 - \sqrt{2}) > 0 \]

### Step 6: Factor the left-hand side
Factoring out common terms:
\[ 2 \sin x (\cos x - \sin x) > 2 - \sqrt{2} \]

### Step 7: Analyze the inequality
We need to find the values of \(x\) for which this inequality holds. This involves solving:
1. \(2 \sin x > 2 - \sqrt{2}\)
2. \(\cos x - \sin x > 0\)

### Step 8: Solve each part
1. From \(2 \sin x > 2 - \sqrt{2}\), we get:
\[ \sin x > 1 - \frac{\sqrt{2}}{2} \]
This gives us the range of \(x\) based on the sine function.

2. From \(\cos x - \sin x > 0\), we can rewrite it as:
\[ \cos x > \sin x \]
This occurs in the first quadrant and part of the fourth quadrant.

### Step 9: Combine the results
We need to combine the solutions from both inequalities to find the intervals where both conditions are satisfied.

### Final Solution
The solution will be the intervals of \(x\) that satisfy both conditions derived from the inequalities.

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