Number of solutions of the equation `sin x + cos x-2sqrt(2) sin x cos x=0` for `x in [0, pi]` is

Options

A. 3
B. 0
C. 1
D. 2

Explanation

To solve the equation \( \sin x + \cos x - 2\sqrt{2} \sin x \cos x = 0 \) for \( x \) in the interval \( [0, \pi] \), we can follow these steps:

### Step 1: Rearrange the Equation
Start by rearranging the equation:
\[ \sin x + \cos x = 2\sqrt{2} \sin x \cos x \]

### Step 2: Square Both Sides
Next, square both sides to eliminate the square root:
\[ (\sin x + \cos x)^2 = (2\sqrt{2} \sin x \cos x)^2 \]
This simplifies to:
\[ \sin^2 x + 2 \sin x \cos x + \cos^2 x = 8 \sin^2 x \cos^2 x \]

### Step 3: Use Trigonometric Identities
Using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) and the double angle identity \( \sin 2x = 2 \sin x \cos x \), we can rewrite the equation:
\[ 1 + \sin 2x = 8 \left(\frac{1}{2} \sin 2x\right)^2 \]
This simplifies to:
\[ 1 + \sin 2x = 2 \sin^2 2x \]

### Step 4: Rearrange into Standard Form
Rearranging gives:
\[ 2 \sin^2 2x - \sin 2x - 1 = 0 \]

### Step 5: Factor the Quadratic Equation
Now, factor the quadratic equation:
\[ (2 \sin 2x + 1)(\sin 2x - 1) = 0 \]
This gives us two equations to solve:
1. \( 2 \sin 2x + 1 = 0 \)
2. \( \sin 2x - 1 = 0 \)

### Step 6: Solve Each Equation
**For \( 2 \sin 2x + 1 = 0 \)**:
\[ \sin 2x = -\frac{1}{2} \]
The general solutions for \( \sin \theta = -\frac{1}{2} \) are:
\[ 2x = \frac{7\pi}{6} + 2k\pi \quad \text{and} \quad 2x = \frac{11\pi}{6} + 2k\pi \]
Dividing by 2 gives:
\[ x = \frac{7\pi}{12} + k\pi \quad \text{and} \quad x = \frac{11\pi}{12} + k\pi \]

**For \( \sin 2x - 1 = 0 \)**:
\[ \sin 2x = 1 \]
The general solution for \( \sin \theta = 1 \) is:
\[ 2x = \frac{\pi}{2} + 2k\pi \]
Dividing by 2 gives:
\[ x = \frac{\pi}{4} + k\pi \]

### Step 7: Determine Valid Solutions in the Interval
Now, we check which solutions lie in the interval \( [0, \pi] \):
1. From \( x = \frac{7\pi}{12} \) (valid)
2. From \( x = \frac{11\pi}{12} \) (valid)
3. From \( x = \frac{\pi}{4} \) (valid)

### Conclusion
Thus, the number of solutions for the equation \( \sin x + \cos x - 2\sqrt{2} \sin x \cos x = 0 \) in the interval \( [0, \pi] \) is **3**.