To simplify the given expression [tex]\(\sin^3 \theta + \frac{\sin \theta}{\sec^2 \theta}\)[/tex], we'll break it down step by step.
### Step 1: Understand the given expression
The given expression is:
[tex]\[
\sin^3 \theta + \frac{\sin \theta}{\sec^2 \theta}
\][/tex]
### Step 2: Simplify [tex]\(\frac{\sin \theta}{\sec^2 \theta}\)[/tex]
Recall the trigonometric identities:
1. [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex]
2. [tex]\(\sec^2 \theta = \left(\frac{1}{\cos \theta}\right)^2 = \frac{1}{\cos^2 \theta}\)[/tex]
Using these, we can rewrite [tex]\(\sec^2 \theta\)[/tex]:
[tex]\[
\sec^2 \theta = \frac{1}{\cos^2 \theta}
\][/tex]
Now, replace [tex]\(\sec^2 \theta\)[/tex] in the original expression:
[tex]\[
\frac{\sin \theta}{\sec^2 \theta} = \sin \theta \cdot \cos^2 \theta
\][/tex]
So the expression now becomes:
[tex]\[
\sin^3 \theta + \sin \theta \cdot \cos^2 \theta
\][/tex]
### Step 3: Factor out [tex]\(\sin \theta\)[/tex]
Notice that [tex]\(\sin \theta\)[/tex] is a common factor in both terms:
[tex]\[
\sin^3 \theta + \sin \theta \cdot \cos^2 \theta = \sin \theta (\sin^2 \theta + \cos^2 \theta)
\][/tex]
### Step 4: Use the Pythagorean identity
Recall the Pythagorean identity in trigonometry:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Substitute this identity into the expression:
[tex]\[
\sin \theta (\sin^2 \theta + \cos^2 \theta) = \sin \theta \cdot 1 = \sin \theta
\][/tex]
### Conclusion
The simplified form of the expression [tex]\(\sin^3 \theta + \frac{\sin \theta}{\sec^2 \theta}\)[/tex] is:
[tex]\[
\sin \theta
\][/tex]
Thus, the answer is:
[tex]\(\sin \theta\)[/tex]
