`sin^(3)theta + sin theta - sin theta cos^(2)theta =`

Options

A. 0
B. `sin theta`
C. `sin 2 theta`
D. `2sin^(3)theta`

Explanation

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

### Step 1: Write down the original expression
We start with the expression:
\[ \sin^3 \theta + \sin \theta - \sin \theta \cos^2 \theta \]

### Step 2: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
This allows us to express \( \cos^2 \theta \) in terms of \( \sin^2 \theta \):
\[ \cos^2 \theta = 1 - \sin^2 \theta \]

### Step 3: Substitute \( \cos^2 \theta \) in the expression
Now, we can substitute \( \cos^2 \theta \) in our expression:
\[ \sin^3 \theta + \sin \theta - \sin \theta (1 - \sin^2 \theta) \]

### Step 4: Distribute \( \sin \theta \)
Next, we distribute \( \sin \theta \) in the expression:
\[ \sin^3 \theta + \sin \theta - \sin \theta + \sin \theta \sin^2 \theta \]

### Step 5: Simplify the expression
Notice that \( \sin \theta - \sin \theta \) cancels out:
\[ \sin^3 \theta + \sin \theta \sin^2 \theta \]

### Step 6: Factor out \( \sin \theta \)
We can factor \( \sin \theta \) out from the remaining terms:
\[ \sin \theta (\sin^2 \theta + \sin^2 \theta) = \sin \theta (2 \sin^2 \theta) \]

### Step 7: Final expression
Thus, we have:
\[ 2 \sin^3 \theta \]

### Conclusion
The final simplified expression is:
\[ 2 \sin^3 \theta \]

### Answer
Therefore, the answer is \( 2 \sin^3 \theta \), which corresponds to option four.

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