If `[(1 + cos theta + i sin theta)/( sin theta + i (1 + cos theta))] ^(4) = cos n theta + i sin n theta , ` then n =

Options

A. 2
B. 3
C. 4
D. none

Answer

To solve the problem, we start with the given expression:

\[ \left( \frac{1 + \cos \theta + i \sin \theta}{\sin \theta + i (1 + \cos \theta)} \right)^{4} = \cos n\theta + i \sin n\theta \]

### Step 1: Simplify the numerator and denominator

The numerator is:
\[ 1 + \cos \theta + i \sin \theta \]
We can rewrite \(1 + \cos \theta\) using the identity:
\[ 1 + \cos \theta = 2 \cos^2 \frac{\theta}{2} \]
Thus, the numerator becomes:
\[ 2 \cos^2 \frac{\theta}{2} + i \sin \theta \]

The denominator is:
\[ \sin \theta + i(1 + \cos \theta) = \sin \theta + i(2 \cos^2 \frac{\theta}{2}) \]

### Step 2: Factor out common terms

Now we can factor out \(2 \cos^2 \frac{\theta}{2}\) from the numerator and denominator:
\[ \frac{2 \cos^2 \frac{\theta}{2} + i \sin \theta}{\sin \theta + i(2 \cos^2 \frac{\theta}{2})} \]

### Step 3: Rewrite in exponential form

Using Euler's formula, we can express \(i \sin \theta\) and \(2 \cos^2 \frac{\theta}{2}\) in terms of exponentials:
\[ \sin \theta = 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} \]
Thus, the numerator becomes:
\[ 2 \cos^2 \frac{\theta}{2} + i(2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}) \]
Factoring out \(2 \cos \frac{\theta}{2}\):
\[ 2 \cos \frac{\theta}{2} \left( \cos \frac{\theta}{2} + i \sin \frac{\theta}{2} \right) = 2 \cos \frac{\theta}{2} e^{i \frac{\theta}{2}} \]

Similarly, the denominator becomes:
\[ 2 \cos^2 \frac{\theta}{2} + i(2 \sin \frac{\theta}{2}) = 2 \cos \frac{\theta}{2} e^{-i \frac{\theta}{2}} \]

### Step 4: Simplify the fraction

Now we can simplify:
\[ \frac{2 \cos \frac{\theta}{2} e^{i \frac{\theta}{2}}}{2 \cos \frac{\theta}{2} e^{-i \frac{\theta}{2}}} = e^{i \theta} \]

### Step 5: Raise to the power of 4

Now we raise this to the power of 4:
\[ (e^{i \theta})^4 = e^{i 4\theta} \]

### Step 6: Compare with the right-hand side

The right-hand side is:
\[ \cos n\theta + i \sin n\theta = e^{i n\theta} \]

### Step 7: Equate the exponents

From \(e^{i 4\theta} = e^{i n\theta}\), we can equate the exponents:
\[ n\theta = 4\theta \]

### Step 8: Solve for n

Dividing both sides by \(\theta\) (assuming \(\theta \neq 0\)):
\[ n = 4 \]

Thus, the value of \(n\) is:
\[ \boxed{4} \]