Let `S(theta)` denote the sum of coefficients in the expansion of `(sqrt2-xsintheta+x^(2)costheta)^(2n)`. Maximum value of `S(theta)` is

Options

A. `4^(n)`
B. `8^(n)`
C. `2^(n)`
D. 1

Answer

To find the maximum value of \( S(\theta) \), which denotes the sum of the coefficients in the expansion of \( (\sqrt{2} - x \sin \theta + x^2 \cos \theta)^{2n} \), we can follow these steps:

### Step 1: Understanding the Sum of Coefficients
The sum of the coefficients of a polynomial \( P(x) \) can be found by evaluating \( P(1) \). Therefore, we need to evaluate:
\[ S(\theta) = (\sqrt{2} - \sin \theta + \cos \theta)^{2n} \]

### Step 2: Substitute \( x = 1 \)
Substituting \( x = 1 \) into the expression gives:
\[ S(\theta) = (\sqrt{2} - \sin \theta + \cos \theta)^{2n} \]

### Step 3: Simplifying the Expression
We can combine the terms inside the parentheses:
\[ S(\theta) = (\sqrt{2} + \cos \theta - \sin \theta)^{2n} \]

### Step 4: Finding Maximum Value
To maximize \( S(\theta) \), we need to maximize the expression \( \sqrt{2} + \cos \theta - \sin \theta \).

### Step 5: Using Trigonometric Identities
We can rewrite \( \cos \theta - \sin \theta \) using the angle addition formula:
\[ \cos \theta - \sin \theta = \sqrt{2} \left( \frac{1}{\sqrt{2}} \cos \theta - \frac{1}{\sqrt{2}} \sin \theta \right) = \sqrt{2} \cos \left( \theta + \frac{\pi}{4} \right) \]
Thus,
\[ S(\theta) = \left( \sqrt{2} + \sqrt{2} \cos \left( \theta + \frac{\pi}{4} \right) \right)^{2n} \]

### Step 6: Maximizing the Cosine Function
The maximum value of \( \cos \left( \theta + \frac{\pi}{4} \right) \) is 1. Therefore, the maximum value of \( S(\theta) \) occurs when:
\[ \cos \left( \theta + \frac{\pi}{4} \right) = 1 \]
This gives:
\[ S(\theta) = \left( \sqrt{2} + \sqrt{2} \cdot 1 \right)^{2n} = (2\sqrt{2})^{2n} \]

### Step 7: Final Calculation
Calculating \( (2\sqrt{2})^{2n} \):
\[ (2\sqrt{2})^{2n} = 2^{2n} \cdot (\sqrt{2})^{2n} = 2^{2n} \cdot 2^n = 2^{3n} \]

### Conclusion
Thus, the maximum value of \( S(\theta) \) is:
\[ \boxed{8^n} \]