The value of `2sinA cos^(3)A -2sin^(3)A cosA` is

Options

A. sin4A
B. `1/2sin4A`
C. `1/4 sin4A`
D. `sin2A`

Answer

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

### Step 1: Factor out common terms
We can factor out \( 2\sin A \cos A \) from the expression:
\[ 2\sin A \cos^3 A - 2\sin^3 A \cos A = 2\sin A \cos A (\cos^2 A - \sin^2 A) \]

### Step 2: Use trigonometric identities
We know that:
- \( \cos^2 A - \sin^2 A = \cos 2A \)
- \( 2\sin A \cos A = \sin 2A \)

Using these identities, we can rewrite the expression:
\[ 2\sin A \cos A (\cos^2 A - \sin^2 A) = \sin 2A \cdot \cos 2A \]

### Step 3: Apply the double angle formula
Now, we can use the double angle formula for sine:
\[ \sin 2A \cdot \cos 2A = \frac{1}{2} \cdot \sin 4A \]

### Final Result
Thus, the value of \( 2\sin A \cos^3 A - 2\sin^3 A \cos A \) simplifies to:
\[ \frac{1}{2} \sin 4A \]

### Conclusion
The final answer is:
\[ \frac{1}{2} \sin 4A \]