`sin4theta` can be written as

Options

A. `4sintheta(1-2sin^(2)theta)sqrt(1-sin^(2)theta)`
B. `2sinthetacosthetasin^(2)theta`
C. `4sintheta-6sin^(3)theta`
D. None of the above

Explanation

To express \( \sin 4\theta \) in terms of \( \sin \theta \), we can follow these steps:

### Step 1: Use the double angle formula for sine
We start with the identity for \( \sin 4\theta \):
\[ \sin 4\theta = \sin(2 \cdot 2\theta) = 2 \sin 2\theta \cos 2\theta \]

### Step 2: Substitute \( \sin 2\theta \)
Next, we apply the double angle formula for \( \sin 2\theta \):
\[ \sin 2\theta = 2 \sin \theta \cos \theta \]
Substituting this into our expression gives:
\[ \sin 4\theta = 2(2 \sin \theta \cos \theta) \cos 2\theta = 4 \sin \theta \cos \theta \cos 2\theta \]

### Step 3: Substitute \( \cos 2\theta \)
Now, we use the double angle formula for cosine:
\[ \cos 2\theta = 1 - 2 \sin^2 \theta \]
Substituting this into our expression results in:
\[ \sin 4\theta = 4 \sin \theta \cos \theta (1 - 2 \sin^2 \theta) \]

### Step 4: Express \( \cos \theta \)
We can express \( \cos \theta \) in terms of \( \sin \theta \) using the Pythagorean identity:
\[ \cos \theta = \sqrt{1 - \sin^2 \theta} \]
Substituting this into our expression gives:
\[ \sin 4\theta = 4 \sin \theta \sqrt{1 - \sin^2 \theta} (1 - 2 \sin^2 \theta) \]

### Final Expression
Thus, we can write:
\[ \sin 4\theta = 4 \sin \theta (1 - 2 \sin^2 \theta) \sqrt{1 - \sin^2 \theta} \]