`4 sin theta * cos^(3) theta-4 cos theta * sin ^(3)theta=` A)`4 cos theta`B)`cos 4 theta`C)`4 sin theta`D)`sin 4 theta`

Options

A. `4 cos theta`
B. `cos 4 theta`
C. `4 sin theta`
D. `sin 4 theta`

Answer

To solve the expression \( 4 \sin \theta \cos^3 \theta - 4 \cos \theta \sin^3 \theta \), we can follow these steps:

### Step 1: Factor out the common terms
We start by factoring out the common terms from the expression. Both terms have a factor of \( 4 \sin \theta \cos \theta \).

\[ 4 \sin \theta \cos^3 \theta - 4 \cos \theta \sin^3 \theta = 4 \sin \theta \cos \theta (\cos^2 \theta - \sin^2 \theta) \]

### Step 2: Use the identity for \( \cos^2 \theta - \sin^2 \theta \)
We know from trigonometric identities that:

\[ \cos^2 \theta - \sin^2 \theta = \cos 2\theta \]

Substituting this into our expression gives:

\[ 4 \sin \theta \cos \theta \cdot \cos 2\theta \]

### Step 3: Use the identity for \( \sin 2\theta \)
We also know that:

\[ \sin 2\theta = 2 \sin \theta \cos \theta \]

Thus, we can express \( 4 \sin \theta \cos \theta \) as:

\[ 4 \sin \theta \cos \theta = 2 \cdot 2 \sin \theta \cos \theta = 2 \sin 2\theta \]

### Step 4: Substitute back into the expression
Now we can substitute this back into our expression:

\[ 4 \sin \theta \cos \theta \cdot \cos 2\theta = 2 \sin 2\theta \cdot \cos 2\theta \]

### Step 5: Use the identity for \( \sin 4\theta \)
We know from the double angle formulas that:

\[ \sin 4\theta = 2 \sin 2\theta \cos 2\theta \]

Thus, we can write:

\[ 2 \sin 2\theta \cos 2\theta = \sin 4\theta \]

### Final Result
Putting it all together, we find that:

\[ 4 \sin \theta \cos^3 \theta - 4 \cos \theta \sin^3 \theta = \sin 4\theta \]

### Conclusion
The final answer is:

\[ \sin 4\theta \]