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 will follow these steps:

### Step 1: Rationalize the Denominator
We will multiply the numerator and the denominator by the conjugate of the denominator, which is \(1 + 2i \cos \theta\):

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

### Step 2: Simplify the Denominator
Using the difference of squares formula, we can simplify the denominator:

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

### Step 3: Simplify the Numerator
Now, we simplify 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 4: Combine the Results
Now we can write the expression as:

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

### Step 5: Separate Real and Imaginary Parts
This expression can be separated into real and imaginary parts:

Real part: \(\frac{1 - 2 \cos^2 \theta}{1 + 4 \cos^2 \theta}\)

Imaginary part: \(\frac{3 \cos \theta}{1 + 4 \cos^2 \theta}\)

### Step 6: Set the Imaginary Part to Zero
For the entire expression to be real, the imaginary part must equal zero:

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

### Step 7: Solve for \(\cos \theta\)
The fraction equals zero when the numerator is zero:

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

### Step 8: Find Values of \(\theta\)
The cosine function equals zero at:

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

This gives us the general solution:

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

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

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