`cos^(2)A+cos^(2)(B-A)-2cosAcosBcos(A-B)=`

Options

A. `cos2A`
B. `sin^(2)A`
C. `sin^(2)B`
D. `cos^(2)B`

Answer

To solve the expression \( \cos^2 A + \cos^2 (B - A) - 2 \cos A \cos B \cos (A - B) \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \cos^2 A + \cos^2 (B - A) - 2 \cos A \cos B \cos (A - B) \]

### Step 2: Use the cosine subtraction formula
Recall the cosine subtraction formula:
\[ \cos (B - A) = \cos B \cos A + \sin B \sin A \]
We can express \( \cos^2 (B - A) \) using the identity:
\[ \cos^2 (B - A) = \cos^2 B \cos^2 A + \sin^2 B \sin^2 A + 2 \cos B \cos A \sin B \sin A \]

### Step 3: Substitute the cosine subtraction formula
Substituting \( \cos^2 (B - A) \) into the expression:
\[ \cos^2 A + \left( \cos^2 B \cos^2 A + \sin^2 B \sin^2 A + 2 \cos B \cos A \sin B \sin A \right) - 2 \cos A \cos B \cos (A - B) \]

### Step 4: Simplify the expression
Now we simplify:
\[ \cos^2 A + \cos^2 B \cos^2 A + \sin^2 B \sin^2 A + 2 \cos B \cos A \sin B \sin A - 2 \cos A \cos B \cos (A - B) \]

### Step 5: Combine like terms
Notice that \( -2 \cos A \cos B \cos (A - B) \) can be simplified using the cosine addition formula:
\[ \cos (A - B) = \cos A \cos B + \sin A \sin B \]
Thus, we can rewrite:
\[ -2 \cos A \cos B (\cos A \cos B + \sin A \sin B) \]

### Step 6: Final simplification
After substituting and simplifying, we can see that the terms will reduce down to:
\[ \cos^2 A + \cos^2 B - \cos^2 A - \cos^2 B = 0 \]

### Final Answer
Thus, the expression simplifies to:
\[ \cos^2 A + \cos^2 (B - A) - 2 \cos A \cos B \cos (A - B) = 0 \]