`f(alpha,beta) = cos^2(alpha)+ cos^2(alpha+beta)- 2 cosalpha cosbeta cos(alpha+beta)` is

Options

A. Is independent of `beta`
B. Is independent of `alpha`
C. Is independent of both `alpha` and `beta`
D. Is dependent of `beta`

Answer

To solve the function \( f(\alpha, \beta) = \cos^2(\alpha) + \cos^2(\alpha + \beta) - 2 \cos(\alpha) \cos(\beta) \cos(\alpha + \beta) \), we will simplify it step by step.

### Step 1: Rewrite the function
We start with the given function:
\[ f(\alpha, \beta) = \cos^2(\alpha) + \cos^2(\alpha + \beta) - 2 \cos(\alpha) \cos(\beta) \cos(\alpha + \beta) \]

### Step 2: Factor out \(\cos(\alpha + \beta)\)
We can factor out \(\cos(\alpha + \beta)\) from the last two terms:
\[ f(\alpha, \beta) = \cos^2(\alpha) + \cos(\alpha + \beta) \left( \cos(\alpha + \beta) - 2 \cos(\alpha) \cos(\beta) \right) \]

### Step 3: Use the cosine addition formula
Using the cosine addition formula, we know that:
\[ \cos(\alpha + \beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta) \]
Substituting this into the function:
\[ f(\alpha, \beta) = \cos^2(\alpha) + \left( \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta) \right) \left( \cos(\alpha + \beta) - 2 \cos(\alpha) \cos(\beta) \right) \]

### Step 4: Simplify the expression
Now, we simplify the expression:
\[ f(\alpha, \beta) = \cos^2(\alpha) + \left( \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta) \right) \left( \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta) - 2 \cos(\alpha) \cos(\beta) \right) \]

### Step 5: Combine like terms
After simplification, we can combine like terms:
\[ f(\alpha, \beta) = \cos^2(\alpha) - \cos^2(\alpha + \beta) + \sin^2(\beta) \]

### Step 6: Final simplification
Finally, we can see that:
\[ f(\alpha, \beta) = \sin^2(\beta) \]

### Conclusion
From the final expression \( f(\alpha, \beta) = \sin^2(\beta) \), we can conclude that the function is independent of \(\alpha\) and dependent on \(\beta\).