`2sinAcos^(3)A-2sin^(3)AcosA=?`

Options

A. `sin4A`
B. `(1)/(2)sin4A`
C. `(1)/(4)sin4A`
D. `(1)/(8)sin4A`

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 the common terms
We notice that both terms in the expression have a common factor of \(2\sin A \cos A\). So, we can factor this out:

\[ 2\sin A \cos^3 A - 2\sin^3 A \cos A = 2\sin A \cos A (\cos^2 A - \sin^2 A) \]

### Step 2: Use the Pythagorean identity
Recall the Pythagorean identity, which states that \(\cos^2 A + \sin^2 A = 1\). We can rewrite \(\cos^2 A - \sin^2 A\) using this identity:

\[ \cos^2 A - \sin^2 A = (\cos^2 A + \sin^2 A) - 2\sin^2 A = 1 - 2\sin^2 A \]

### Step 3: Substitute back into the expression
Now we substitute this back into our factored expression:

\[ 2\sin A \cos A (1 - 2\sin^2 A) \]

### Step 4: Recognize the double angle formula
The term \(2\sin A \cos A\) can be recognized as the double angle formula for sine:

\[ 2\sin A \cos A = \sin(2A) \]

### Step 5: Final expression
Thus, we can rewrite the entire expression as:

\[ \sin(2A)(1 - 2\sin^2 A) \]

### Conclusion
The final result of the expression \(2\sin A \cos^3 A - 2\sin^3 A \cos A\) is:

\[ \sin(2A)(1 - 2\sin^2 A) \]

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