Find the values of the following expression. `sin^4theta+sin^2thetacos^2theta`=?

Options

A. 0
B. 1
C. 2
D. `sin^2theta`

Answer

To solve the expression \( \sin^4 \theta + \sin^2 \theta \cos^2 \theta \), we can follow these steps:

### Step 1: Factor out \( \sin^2 \theta \)
We start by factoring out \( \sin^2 \theta \) from the expression:

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

### Step 2: Use the Pythagorean Identity
Next, we apply the Pythagorean identity, which states that \( \sin^2 \theta + \cos^2 \theta = 1 \):

\[ \sin^2 \theta (\sin^2 \theta + \cos^2 \theta) = \sin^2 \theta \cdot 1 \]

### Step 3: Simplify the Expression
Now we can simplify the expression:

\[ \sin^2 \theta \cdot 1 = \sin^2 \theta \]

### Conclusion
Thus, the value of the expression \( \sin^4 \theta + \sin^2 \theta \cos^2 \theta \) is:

\[ \sin^2 \theta \]

### Final Answer
The final answer is \( \sin^2 \theta \).

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