The real value of `theta` for which the expression `(1 + icos theta)/(1 - 2i cos theta)` is real number is

Options

A. `npi+(pi)/(4)`
B. `npi+(-1)^(2)(pi)/(4)`
C. `2npipm(pi)/(2)`
D. None of these

Explanation

To find the real value of \( \theta \) for which the expression

\[ \frac{1 + i \cos \theta}{1 - 2i \cos \theta} \]

is a real number, we can follow these steps:

### Step 1: Identify the expression
We start with the expression:

\[ z = \frac{1 + i \cos \theta}{1 - 2i \cos \theta} \]

### Step 2: Rationalize the denominator
To simplify this expression and check when it is real, we can multiply the numerator and the denominator by the conjugate of the denominator:

\[ z = \frac{(1 + i \cos \theta)(1 + 2i \cos \theta)}{(1 - 2i \cos \theta)(1 + 2i \cos \theta)} \]

### Step 3: Simplify the denominator
The denominator simplifies as follows:

\[ (1 - 2i \cos \theta)(1 + 2i \cos \theta) = 1^2 - (2i \cos \theta)^2 = 1 - 4(-1)(\cos^2 \theta) = 1 + 4 \cos^2 \theta \]

### Step 4: Simplify the numerator
Now simplifying the numerator:

\[ (1 + i \cos \theta)(1 + 2i \cos \theta) = 1 + 2i \cos \theta + i \cos \theta + 2i^2 \cos^2 \theta = 1 + 3i \cos \theta - 2 \cos^2 \theta \]

### Step 5: Combine the results
Now we can combine the results:

\[ z = \frac{1 - 2 \cos^2 \theta + 3i \cos \theta}{1 + 4 \cos^2 \theta} \]

### Step 6: Separate real and imaginary parts
The real part of \( z \) is:

\[ \text{Re}(z) = \frac{1 - 2 \cos^2 \theta}{1 + 4 \cos^2 \theta} \]

The imaginary part of \( z \) is:

\[ \text{Im}(z) = \frac{3 \cos \theta}{1 + 4 \cos^2 \theta} \]

### Step 7: Set the imaginary part to zero
For \( z \) to be a real number, the imaginary part must equal zero:

\[ \frac{3 \cos \theta}{1 + 4 \cos^2 \theta} = 0 \]

### Step 8: Solve for \( \cos \theta \)
This implies:

\[ 3 \cos \theta = 0 \implies \cos \theta = 0 \]

### Step 9: Find the values of \( \theta \)
The values of \( \theta \) for which \( \cos \theta = 0 \) are:

\[ \theta = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]

This can be expressed as:

\[ \theta = 2n\pi + \frac{\pi}{2} \quad \text{or} \quad \theta = 2n\pi - \frac{\pi}{2} \]

### Final Answer
Thus, the real values of \( \theta \) for which the expression is a real number are:

\[ \theta = 2n\pi + \frac{\pi}{2} \quad \text{for } n \in \mathbb{Z} \]