The values of x between 0 and `2pi` which satisfy the equation `sinxsqrt(8 cos^2 x)= 1` are in A.P. with common difference is

Options

A. ` (pi)/(4) `
B. `(pi)/(3) `
C. `(pi)/(2)`
D. `(2 pi)/(3)`

Answer

To solve the equation \( \sin x \sqrt{8 \cos^2 x} = 1 \) and find the values of \( x \) between \( 0 \) and \( 2\pi \) that satisfy this equation, we will follow these steps:

### Step 1: Simplify the Equation
Start with the given equation:
\[ \sin x \sqrt{8 \cos^2 x} = 1 \]
We can rewrite \( \sqrt{8 \cos^2 x} \) as \( 2\sqrt{2} \cos x \):
\[ \sin x \cdot 2\sqrt{2} \cos x = 1 \]
This simplifies to:
\[ 2\sqrt{2} \sin x \cos x = 1 \]

### Step 2: Use the Double Angle Identity
Using the double angle identity, \( \sin 2x = 2 \sin x \cos x \), we can rewrite the equation:
\[ \sqrt{2} \sin 2x = 1 \]
Dividing both sides by \( \sqrt{2} \):
\[ \sin 2x = \frac{1}{\sqrt{2}} \]

### Step 3: Solve for \( 2x \)
The general solution for \( \sin \theta = \frac{1}{\sqrt{2}} \) is:
\[ \theta = \frac{\pi}{4} + 2k\pi \quad \text{or} \quad \theta = \frac{3\pi}{4} + 2k\pi \]
Setting \( \theta = 2x \), we have:
1. \( 2x = \frac{\pi}{4} + 2k\pi \)
2. \( 2x = \frac{3\pi}{4} + 2k\pi \)

### Step 4: Solve for \( x \)
Dividing by 2 gives:
1. \( x = \frac{\pi}{8} + k\pi \)
2. \( x = \frac{3\pi}{8} + k\pi \)

### Step 5: Find Values of \( x \) in the Interval \( [0, 2\pi] \)
Now we will find values of \( x \) for \( k = 0 \) and \( k = 1 \):
- For \( k = 0 \):
- From \( x = \frac{\pi}{8} \) → \( x = \frac{\pi}{8} \)
- From \( x = \frac{3\pi}{8} \) → \( x = \frac{3\pi}{8} \)

- For \( k = 1 \):
- From \( x = \frac{\pi}{8} + \pi = \frac{9\pi}{8} \)
- From \( x = \frac{3\pi}{8} + \pi = \frac{11\pi}{8} \)

### Step 6: List All Values of \( x \)
The values of \( x \) that satisfy the equation in the interval \( [0, 2\pi] \) are:
- \( x = \frac{\pi}{8} \)
- \( x = \frac{3\pi}{8} \)
- \( x = \frac{9\pi}{8} \)
- \( x = \frac{11\pi}{8} \)

### Step 7: Check if the Values are in A.P.
To check if these values are in arithmetic progression (A.P.), we can calculate the common difference:
- The first term \( a_1 = \frac{\pi}{8} \)
- The second term \( a_2 = \frac{3\pi}{8} \)
- The third term \( a_3 = \frac{9\pi}{8} \)
- The fourth term \( a_4 = \frac{11\pi}{8} \)

Calculating the common differences:
- \( a_2 - a_1 = \frac{3\pi}{8} - \frac{\pi}{8} = \frac{2\pi}{8} = \frac{\pi}{4} \)
- \( a_3 - a_2 = \frac{9\pi}{8} - \frac{3\pi}{8} = \frac{6\pi}{8} = \frac{3\pi}{4} \)
- \( a_4 - a_3 = \frac{11\pi}{8} - \frac{9\pi}{8} = \frac{2\pi}{8} = \frac{\pi}{4} \)

### Conclusion
The common difference of the A.P. is:
\[ \frac{\pi}{4} \]