The complex number `sin(x)+icos(2x)` and `cos(x)-isin(2x)` are conjugate to each other for

Options

A. `x = npi, n in Z`
B. `x= 0`
C. `x = (n+1//2)pi, n in Z`
D. no value of x

Explanation

To determine the values of \( x \) for which the complex numbers \( \sin(x) + i \cos(2x) \) and \( \cos(x) - i \sin(2x) \) are conjugates of each other, we need to set up the equality between the two expressions and analyze their real and imaginary parts.

### Step 1: Set the two complex numbers equal to each other.
We have:
\[ \sin(x) + i \cos(2x) = \cos(x) - i \sin(2x) \]

### Step 2: Separate the real and imaginary parts.
From the above equation, we can equate the real parts and the imaginary parts:
- Real part:
\[ \sin(x) = \cos(x) \]
- Imaginary part:
\[ \cos(2x) = -\sin(2x) \]

### Step 3: Solve the real part equation.
From the equation \( \sin(x) = \cos(x) \), we can divide both sides by \( \cos(x) \) (assuming \( \cos(x) \neq 0 \)):
\[ \tan(x) = 1 \]
This implies:
\[ x = \frac{\pi}{4} + n\pi \quad (n \in \mathbb{Z}) \]

### Step 4: Solve the imaginary part equation.
From the equation \( \cos(2x) = -\sin(2x) \), we can rewrite it as:
\[ \cos(2x) + \sin(2x) = 0 \]
Dividing by \( \cos(2x) \) (assuming \( \cos(2x) \neq 0 \)):
\[ 1 + \tan(2x) = 0 \implies \tan(2x) = -1 \]
This implies:
\[ 2x = \frac{3\pi}{4} + n\pi \quad (n \in \mathbb{Z}) \]
Thus:
\[ x = \frac{3\pi}{8} + \frac{n\pi}{2} \quad (n \in \mathbb{Z}) \]

### Step 5: Find common solutions.
We need to find values of \( x \) that satisfy both:
1. \( x = \frac{\pi}{4} + n\pi \)
2. \( x = \frac{3\pi}{8} + \frac{n\pi}{2} \)

To find common solutions, we can equate:
\[ \frac{\pi}{4} + n\pi = \frac{3\pi}{8} + \frac{m\pi}{2} \]
Multiply through by 8 to eliminate the fractions:
\[ 2\pi + 8n\pi = 3\pi + 4m\pi \]
Rearranging gives:
\[ 8n\pi - 4m\pi = \pi \implies 8n - 4m = 1 \]
This equation does not yield integer solutions for \( n \) and \( m \), indicating that there are no common solutions.

### Conclusion:
The complex numbers \( \sin(x) + i \cos(2x) \) and \( \cos(x) - i \sin(2x) \) are conjugates for no values of \( x \).

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