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

Options

A. `cos2A`
B. `8cos2A`
C. `1//8cos2A`
D. none of these

Answer

To solve the expression \(\frac{\sin^2 3A}{\sin^2 A} - \frac{\cos^2 3A}{\cos^2 A}\), we will follow these steps:

### Step 1: Rewrite the expression
Start by rewriting the expression using a common denominator:
\[ \frac{\sin^2 3A \cdot \cos^2 A - \cos^2 3A \cdot \sin^2 A}{\sin^2 A \cdot \cos^2 A} \]

### Step 2: Recognize the difference of squares
Notice that the numerator can be expressed as a difference of squares:
\[ \sin^2 3A \cdot \cos^2 A - \cos^2 3A \cdot \sin^2 A = (\sin 3A \cdot \cos A + \sin A \cdot \cos 3A)(\sin 3A \cdot \cos A - \sin A \cdot \cos 3A) \]

### Step 3: Apply the sine addition and subtraction formulas
Using the sine addition and subtraction formulas:
- \(\sin(a + b) = \sin a \cos b + \cos a \sin b\)
- \(\sin(a - b) = \sin a \cos b - \cos a \sin b\)

We can rewrite the factors:
1. \(\sin(3A + A) = \sin(4A)\)
2. \(\sin(3A - A) = \sin(2A)\)

Thus, we can rewrite the numerator:
\[ \sin 4A \cdot \sin 2A \]

### Step 4: Substitute back into the expression
Now substitute back into the expression:
\[ \frac{\sin 4A \cdot \sin 2A}{\sin^2 A \cdot \cos^2 A} \]

### Step 5: Simplify using the double angle formula
Recall that \(\sin 4A = 2 \sin 2A \cos 2A\). Therefore:
\[ \frac{2 \sin 2A \cos 2A \cdot \sin 2A}{\sin^2 A \cdot \cos^2 A} \]

### Step 6: Combine the terms
This simplifies to:
\[ \frac{2 \sin^2 2A \cos 2A}{\sin^2 A \cdot \cos^2 A} \]

### Step 7: Factor out constants
Now, we can multiply and divide by 4:
\[ \frac{4 \sin^2 2A \cos 2A}{4 \sin^2 A \cos^2 A} \]

### Step 8: Cancel out \(\sin^2 2A\)
Since \(\sin^2 2A\) and \(\sin^2 A\) are both present, we can simplify:
\[ = 8 \cos 2A \]

### Final Answer
Thus, the final answer is:
\[ 8 \cos 2A \]

---