Question

What is a simplified form of the expression sine cubed theta plus Start Fraction sine theta over secant squared theta End Fraction?
(1 point)
Responses

csc θ
csc θ

sin θ
sin θ

sin θ + cosθ
sin θ + cos θ

2 sine cubed theta

Answer 1

To simplify the expression \( \sin^3 \theta + \frac{\sin \theta}{\sec^2 \theta} \), we start by noting that \( \sec \theta = \frac{1}{\cos \theta} \), so:

\[
\sec^2 \theta = \frac{1}{\cos^2 \theta}
\]

Therefore, we can rewrite \( \frac{\sin \theta}{\sec^2 \theta} \) as follows:

\[
\frac{\sin \theta}{\sec^2 \theta} = \sin \theta \cdot \cos^2 \theta
\]

Now substituting back into the expression:

\[
\sin^3 \theta + \sin \theta \cdot \cos^2 \theta
\]

Next, we can factor out \( \sin \theta \):

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

From the Pythagorean identity, we know that:

\[
\sin^2 \theta + \cos^2 \theta = 1
\]

So the expression simplifies to:

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

Thus, the simplified form of the expression is:

\[
\sin \theta
\]

The correct response is:

**sin θ**