`(sin^(2)3A)/(sin^2A)-(cos^(2)3A)/(cos^2A)=`

Options

A. `cos2A`
B. `8cos2A`
C. `1/8cos2A`
D. `1/4cos2A`

Answer

To solve the expression \(\frac{\sin^2 3A}{\sin^2 A} - \frac{\cos^2 3A}{\cos^2 A}\), we can use the trigonometric identities for \(\sin 3A\) and \(\cos 3A\).

### Step 1: Use the triple angle identities
We know the following identities:
\[ \sin 3A = 3\sin A - 4\sin^3 A \]
\[ \cos 3A = 4\cos^3 A - 3\cos A \]

### Step 2: Substitute the identities into the expression
Substituting these identities into the expression gives:
\[ \frac{(3\sin A - 4\sin^3 A)^2}{\sin^2 A} - \frac{(4\cos^3 A - 3\cos A)^2}{\cos^2 A} \]

### Step 3: Simplify the first term
For the first term, we can factor out \(\sin^2 A\):
\[ \frac{(3\sin A - 4\sin^3 A)^2}{\sin^2 A} = \frac{(3 - 4\sin^2 A)^2 \sin^2 A}{\sin^2 A} = (3 - 4\sin^2 A)^2 \]

### Step 4: Simplify the second term
For the second term, we can factor out \(\cos^2 A\):
\[ \frac{(4\cos^3 A - 3\cos A)^2}{\cos^2 A} = \frac{(4\cos^2 A - 3)^2 \cos^2 A}{\cos^2 A} = (4\cos^2 A - 3)^2 \]

### Step 5: Combine the simplified terms
Now we can combine the simplified terms:
\[ (3 - 4\sin^2 A)^2 - (4\cos^2 A - 3)^2 \]

### Step 6: Use the difference of squares
This expression can be simplified using the difference of squares:
\[ (a^2 - b^2) = (a - b)(a + b) \]
Let \(a = 3 - 4\sin^2 A\) and \(b = 4\cos^2 A - 3\):
\[ (3 - 4\sin^2 A - (4\cos^2 A - 3))(3 - 4\sin^2 A + (4\cos^2 A - 3)) \]

### Step 7: Simplify both factors
1. **First factor**:
\[ 3 - 4\sin^2 A - 4\cos^2 A + 3 = 6 - 4(\sin^2 A + \cos^2 A) = 6 - 4 = 2 \]
2. **Second factor**:
\[ 3 - 4\sin^2 A + 4\cos^2 A - 3 = -4\sin^2 A + 4\cos^2 A = 4(\cos^2 A - \sin^2 A) \]

### Step 8: Final expression
Thus, the expression simplifies to:
\[ 2 \cdot 4(\cos^2 A - \sin^2 A) = 8(\cos^2 A - \sin^2 A) \]

### Final Answer
The final answer is:
\[ 8(\cos^2 A - \sin^2 A) \]