`sin^(2)Acos^(2)B-cos^(2)A sin^(2)B` simplifies to :

Options

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

Answer

To simplify the expression \( \sin^2 A \cos^2 B - \cos^2 A \sin^2 B \), we can follow these steps:

### Step 1: Rewrite the expression
The given expression is:
\[ \sin^2 A \cos^2 B - \cos^2 A \sin^2 B \]

### Step 2: Use the identity for \( \cos^2 B \)
Recall the Pythagorean identity:
\[ \cos^2 B = 1 - \sin^2 B \]
Substituting this into the expression gives:
\[ \sin^2 A (1 - \sin^2 B) - \cos^2 A \sin^2 B \]

### Step 3: Distribute \( \sin^2 A \)
Distributing \( \sin^2 A \) in the first term:
\[ \sin^2 A - \sin^2 A \sin^2 B - \cos^2 A \sin^2 B \]

### Step 4: Combine like terms
Now, combine the terms involving \( \sin^2 B \):
\[ \sin^2 A - (\sin^2 A + \cos^2 A) \sin^2 B \]

### Step 5: Use the identity \( \sin^2 A + \cos^2 A = 1 \)
Using the identity \( \sin^2 A + \cos^2 A = 1 \):
\[ \sin^2 A - 1 \cdot \sin^2 B \]
This simplifies to:
\[ \sin^2 A - \sin^2 B \]

### Step 6: Final expression
Thus, the simplified expression is:
\[ \sin^2 A - \sin^2 B \]

### Summary
The expression \( \sin^2 A \cos^2 B - \cos^2 A \sin^2 B \) simplifies to \( \sin^2 A - \sin^2 B \).

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