`-cos(A+B)cos(B-A) =`

Options

A. `cos^(2)A - sin^(2)B`
B. `cos^(2)B - sin^(2)A`
C. `sin^(2)B - cos^(2)A`
D. None of these

Answer

To solve the expression \(-\cos(A+B)\cos(B-A)\), we can use the cosine addition and subtraction formulas. Let's break it down step by step.

### Step 1: Apply the Cosine Addition and Subtraction Formulas
The cosine addition and subtraction formulas are:
\[ \cos(A+B) = \cos A \cos B - \sin A \sin B \]
\[ \cos(B-A) = \cos B \cos A + \sin B \sin A \]

### Step 2: Substitute the Formulas into the Expression
Substituting these formulas into the expression, we have:
\[ -\cos(A+B)\cos(B-A) = -(\cos A \cos B - \sin A \sin B)(\cos B \cos A + \sin B \sin A) \]

### Step 3: Expand the Expression
Now, we will expand the expression:
\[ = -[(\cos A \cos B)(\cos B \cos A) + (\cos A \cos B)(\sin B \sin A) - (\sin A \sin B)(\cos B \cos A) - (\sin A \sin B)(\sin B \sin A)] \]
\[ = -[\cos^2 A \cos^2 B + \cos A \cos B \sin B \sin A - \sin A \sin B \cos B \cos A - \sin^2 A \sin^2 B] \]
Notice that the middle terms cancel out:
\[ = -[\cos^2 A \cos^2 B - \sin^2 A \sin^2 B] \]

### Step 4: Factor the Result
We can factor the result as:
\[ = -(\cos^2 A \cos^2 B - \sin^2 A \sin^2 B) = -(\cos^2 A \cos^2 B - \sin^2 A \sin^2 B) \]
This can be rewritten using the identity \(a^2 - b^2 = (a-b)(a+b)\):
\[ = -[(\cos^2 A - \sin^2 A)(\cos^2 B + \sin^2 B)] \]
Since \(\cos^2 B + \sin^2 B = 1\):
\[ = -(\cos^2 A - \sin^2 A) \]

### Step 5: Final Result
Thus, the final result is:
\[ = \sin^2 A - \cos^2 A \]

### Summary of Steps:
1. Apply cosine addition and subtraction formulas.
2. Substitute these formulas into the expression.
3. Expand the expression and simplify.
4. Factor the result using the difference of squares.
5. Conclude with the final simplified expression.