sinx + i cos 2x and cos x - i sin2x are conjugate to each other for (A) x=nπ (B) x=(n+1/2)π/2 (C) x=0 (D) no value of x

Options

A. `x = npi`
B. `x = ("n" +(1)/(2))(pi)/(2)`
C. x=0
D. No value of x

Explanation

To determine for which values of \( x \) the expressions \( \sin x + i \cos 2x \) and \( \cos x - i \sin 2x \) are conjugates of each other, we can follow these steps:

### Step 1: Define the Expressions
Let:
\[ z = \sin x + i \cos 2x \]
The conjugate of \( z \) is:
\[ \overline{z} = \sin x - i \cos 2x \]

### Step 2: Set Up the Conjugate Condition
We want to find when \( \overline{z} = \cos x - i \sin 2x \). This gives us two equations by equating the real and imaginary parts:
1. Real part:
\[ \sin x = \cos x \]
2. Imaginary part:
\[ -\cos 2x = -\sin 2x \quad \text{(which simplifies to)} \quad \cos 2x = \sin 2x \]

### Step 3: Solve the First Equation
From the equation \( \sin x = \cos x \):
\[ \tan x = 1 \]
This implies:
\[ x = \frac{\pi}{4} + n\pi \quad \text{for } n \in \mathbb{Z} \]

### Step 4: Solve the Second Equation
From the equation \( \cos 2x = \sin 2x \):
\[ \tan 2x = 1 \]
This implies:
\[ 2x = \frac{\pi}{4} + m\pi \quad \text{for } m \in \mathbb{Z} \]
Dividing by 2 gives:
\[ x = \frac{\pi}{8} + \frac{m\pi}{2} \]

### Step 5: Find Common Solutions
Now we need to find values of \( x \) that satisfy both equations:
1. From \( x = \frac{\pi}{4} + n\pi \)
2. From \( x = \frac{\pi}{8} + \frac{m\pi}{2} \)

To find common solutions, we can set:
\[ \frac{\pi}{4} + n\pi = \frac{\pi}{8} + \frac{m\pi}{2} \]
Multiplying through by 8 to eliminate the fractions:
\[ 2\pi + 8n\pi = \pi + 4m\pi \]
Rearranging gives:
\[ 8n\pi - 4m\pi = -\pi \]
This simplifies to:
\[ 8n - 4m = -1 \]
or
\[ 8n + 1 = 4m \]

### Step 6: Check for Integer Solutions
Since \( m \) must be an integer, \( 8n + 1 \) must be divisible by 4. This means \( 8n + 1 \equiv 0 \mod 4 \), which simplifies to \( 1 \equiv 0 \mod 4 \), indicating that there are no integer solutions for \( n \) and \( m \).

### Conclusion
Thus, the two expressions are conjugate to each other for no values of \( x \).

The correct answer is:
**(D) no value of x**