If `f(alpha,beta)=cos^2alpha+sin^2alphacos2beta` then which of the following is incorrect

Options

A. `f(pi/5, (2pi)/5) ne f((2pi)/5, pi/6)`
B. `f(pi/12, pi/3)= f(pi/3, pi/12)`
C. `3f(pi/5, pi/3) ne f(pi/3, pi/5)`
D. `f(pi/4, pi/18) ne 3f(pi/18, pi/4)`

Answer

To solve the problem, we need to analyze the function \( f(\alpha, \beta) = \cos^2 \alpha + \sin^2 \alpha \cos 2\beta \) and determine which of the given options is incorrect.

### Step-by-Step Solution:

1. **Write down the function:**
\[ f(\alpha, \beta) = \cos^2 \alpha + \sin^2 \alpha \cos 2\beta \]

2. **Use the identity for \(\sin^2 \alpha\):**
Recall that \(\sin^2 \alpha = 1 - \cos^2 \alpha\). Substitute this into the function:
\[ f(\alpha, \beta) = \cos^2 \alpha + (1 - \cos^2 \alpha) \cos 2\beta \]

3. **Expand the function:**
\[ f(\alpha, \beta) = \cos^2 \alpha + \cos 2\beta - \cos^2 \alpha \cos 2\beta \]

4. **Factor the expression:**
Combine the terms:
\[ f(\alpha, \beta) = \cos^2 \alpha (1 - \cos 2\beta) + \cos 2\beta \]

5. **Use the identity for \(\cos 2\beta\):**
Recall that \(\cos 2\beta = 2\cos^2 \beta - 1\). Substitute this into the function:
\[ f(\alpha, \beta) = \cos^2 \alpha (1 - (2\cos^2 \beta - 1)) + (2\cos^2 \beta - 1) \]
Simplifying gives:
\[ f(\alpha, \beta) = \cos^2 \alpha (2 - 2\cos^2 \beta) + (2\cos^2 \beta - 1) \]

6. **Distribute and combine like terms:**
\[ f(\alpha, \beta) = 2\cos^2 \alpha - 2\cos^2 \alpha \cos^2 \beta + 2\cos^2 \beta - 1 \]

7. **Rearranging the function:**
\[ f(\alpha, \beta) = 2\cos^2 \beta + 2\cos^2 \alpha - 2\cos^2 \alpha \cos^2 \beta - 1 \]

8. **Check for symmetry:**
We can check if \( f(\alpha, \beta) = f(\beta, \alpha) \):
\[ f(\beta, \alpha) = 2\cos^2 \alpha + 2\cos^2 \beta - 2\cos^2 \beta \cos^2 \alpha - 1 \]
This shows that \( f(\alpha, \beta) = f(\beta, \alpha) \), confirming the function is symmetric.

9. **Evaluate the options:**
We need to determine which option is incorrect based on the symmetry we found. The incorrect option will not satisfy \( f(\alpha, \beta) = f(\beta, \alpha) \).

### Conclusion:
After evaluating the options, we find that the incorrect option is \( f(\frac{\pi}{5}, 2) \neq f(2, \frac{\pi}{5}) \).