The solutions of the equation `1+(sin x - cos x)"sin" pi/4=2 "cos"^(2) (5 x)/2` is/are

Options

A. `x=(n pi)/3+pi/8, n in Z`
B. `x=(npi)/2+(5pi)/16, n in Z`
C. `x=(npi)/3+pi/4, n in Z`
D. `x=(npi)/2+(7pi)/8, n in Z`

Explanation

To solve the equation \( 1 + (\sin x - \cos x) \sin \frac{\pi}{4} = 2 \cos^2 \frac{5x}{2} \), we will follow these steps:

### Step 1: Simplify the left-hand side
We know that \( \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \). Thus, we can rewrite the equation as:
\[ 1 + (\sin x - \cos x) \cdot \frac{1}{\sqrt{2}} = 2 \cos^2 \frac{5x}{2} \]

### Step 2: Rewrite the equation
This simplifies to:
\[ 1 + \frac{\sin x - \cos x}{\sqrt{2}} = 2 \cos^2 \frac{5x}{2} \]

### Step 3: Rearranging the equation
Subtract 1 from both sides:
\[ \frac{\sin x - \cos x}{\sqrt{2}} = 2 \cos^2 \frac{5x}{2} - 1 \]

### Step 4: Use the identity for cosine
We know that \( 2 \cos^2 \theta - 1 = \cos(2\theta) \). Therefore, we can rewrite the right-hand side:
\[ \frac{\sin x - \cos x}{\sqrt{2}} = \cos(5x) \]

### Step 5: Multiply both sides by \(\sqrt{2}\)
This gives us:
\[ \sin x - \cos x = \sqrt{2} \cos(5x) \]

### Step 6: Use the identity for sine and cosine
We can rewrite \( \sin x - \cos x \) as:
\[ \sqrt{2} \sin\left(x - \frac{\pi}{4}\right) \]
Thus, we have:
\[ \sqrt{2} \sin\left(x - \frac{\pi}{4}\right) = \sqrt{2} \cos(5x) \]

### Step 7: Divide by \(\sqrt{2}\)
This simplifies to:
\[ \sin\left(x - \frac{\pi}{4}\right) = \cos(5x) \]

### Step 8: Use the co-function identity
Using the identity \( \sin A = \cos B \) gives us:
\[ x - \frac{\pi}{4} = 5x + n\pi \quad \text{or} \quad x - \frac{\pi}{4} = -5x + n\pi \]

### Step 9: Solve the first equation
From \( x - \frac{\pi}{4} = 5x + n\pi \):
\[ -\frac{4x}{4} = n\pi + \frac{\pi}{4} \implies x = -\frac{n\pi + \frac{\pi}{4}}{4} \]

### Step 10: Solve the second equation
From \( x - \frac{\pi}{4} = -5x + n\pi \):
\[ 6x = n\pi + \frac{\pi}{4} \implies x = \frac{n\pi + \frac{\pi}{4}}{6} \]

### Step 11: Combine solutions
Thus, we have two families of solutions:
1. \( x = -\frac{n\pi + \frac{\pi}{4}}{4} \)
2. \( x = \frac{n\pi + \frac{\pi}{4}}{6} \)

### Conclusion
The solutions of the equation are:
- \( x = n\frac{\pi}{3} + \frac{\pi}{8} \) (Option 1)
- \( x = n\frac{\pi}{2} + \frac{5\pi}{16} \) (Option 2)
- \( x = n\frac{\pi}{3} + \frac{\pi}{4} \) (Option 3)
- \( x = n\frac{\pi}{2} + \frac{7\pi}{8} \) (Option 4)

The correct option is Option 2: \( x = n\frac{\pi}{2} + \frac{5\pi}{16} \).