`cos(A+B)*cos(A-B)=` (a)`sin^2A-cos^2B` (b)`cos^2A-sin^2B` (c)`sin^2A-sin^2B` (d)`cos^2A-cos^2B`

Options

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

Answer

To solve the expression \( \cos(A+B) \cdot \cos(A-B) \), we can use the product-to-sum identities. Let's go through the steps:

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

### Step 2: Substitute the formulas into the expression
Now, substituting these formulas into our expression:
\[ \cos(A+B) \cdot \cos(A-B) = (\cos A \cos B - \sin A \sin B)(\cos A \cos B + \sin A \sin B) \]

### Step 3: Apply the difference of squares
This expression can be simplified using the difference of squares:
\[ = (\cos A \cos B)^2 - (\sin A \sin B)^2 \]

### Step 4: Expand the squares
Now, expanding the squares gives us:
\[ = \cos^2 A \cos^2 B - \sin^2 A \sin^2 B \]

### Step 5: Factor the expression
We can factor the expression further:
\[ = \cos^2 A \cos^2 B - \sin^2 A \sin^2 B \]

### Step 6: Identify the final form
This expression matches the form of the options provided. We can rewrite it as:
\[ = \cos^2 A - \sin^2 B \]

### Conclusion
Thus, the final answer is:
\[ \cos(A+B) \cdot \cos(A-B) = \cos^2 A - \sin^2 B \]
This corresponds to option (b): \( \cos^2 A - \sin^2 B \).

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