If `((1 + cos theta + i sin theta ) /(i+ sin theta + i cos theta ))= cos ntheta + i sin n theta ` then n is equal to :

Options

A. 2
B. 4
C. 6
D. 8

Answer

To solve the given problem, we need to simplify the expression and find the value of \( n \) such that:

\[ \frac{1 + \cos \theta + i \sin \theta}{i + \sin \theta + i \cos \theta} = \cos n\theta + i \sin n\theta \]

### Step-by-Step Solution:

1. **Simplify the Numerator:**
The numerator is \( 1 + \cos \theta + i \sin \theta \). We can rewrite this as:
\[ 1 + \cos \theta + i \sin \theta = 1 + \cos \theta + i \sin \theta \]

2. **Simplify the Denominator:**
The denominator is \( i + \sin \theta + i \cos \theta \). We can factor out \( i \):
\[ i(1 + \cos \theta) + \sin \theta \]

3. **Combine the Expression:**
Now, we can rewrite the entire expression:
\[ \frac{1 + \cos \theta + i \sin \theta}{i(1 + \cos \theta) + \sin \theta} \]

4. **Multiply by the Conjugate:**
To simplify further, we multiply the numerator and denominator by the conjugate of the denominator:
\[ \text{Conjugate of } (i + \sin \theta + i \cos \theta) = -i + \sin \theta - i \cos \theta \]

5. **Calculate the Denominator:**
The denominator becomes:
\[ (i + \sin \theta + i \cos \theta)(-i + \sin \theta - i \cos \theta) \]
This simplifies to:
\[ \sin^2 \theta + 1 + \cos^2 \theta = 2 \]

6. **Calculate the Numerator:**
The numerator becomes:
\[ (1 + \cos \theta + i \sin \theta)(-i + \sin \theta - i \cos \theta) \]
Expanding this gives:
\[ (1 + \cos \theta)(-i) + (1 + \cos \theta)(\sin \theta) + i \sin \theta(-i) + i \sin \theta(-i \cos \theta) \]

7. **Combine Terms:**
After simplification, we will have:
\[ \text{Numerator} = \sin \theta + \cos \theta + i(1 + \cos \theta) \]

8. **Final Expression:**
Now we can write:
\[ \frac{\sin \theta + \cos \theta + i(1 + \cos \theta)}{2} \]

9. **Set Equal to Right Side:**
Now we equate this to \( \cos n\theta + i \sin n\theta \):
\[ \frac{\sin \theta + \cos \theta + i(1 + \cos \theta)}{2} = \cos n\theta + i \sin n\theta \]

10. **Extract Real and Imaginary Parts:**
From the above equation, we can equate the real and imaginary parts:
- Real part: \( \frac{\sin \theta + \cos \theta}{2} = \cos n\theta \)
- Imaginary part: \( \frac{1 + \cos \theta}{2} = \sin n\theta \)

11. **Use Trigonometric Identities:**
We can use the half-angle formulas and properties of sine and cosine to find \( n \).

12. **Final Calculation:**
We find that:
\[ n\theta = \theta - \frac{\pi}{2} \]
Solving for \( n \):
\[ n = 1 - \frac{\pi}{2\theta} \]

### Conclusion:
Thus, the value of \( n \) is:
\[ n = 1 - \frac{\pi}{2\theta} \]