Alright, let's break down the given mathematical expression step-by-step.
Given expression:
[tex]\[
\frac{\left(\frac{\sin \theta}{\sec \theta}\right)}{\left(\frac{\cos \theta}{\sin \theta}\right)}
\][/tex]
### Step 1: Simplify the inner terms
First, let's simplify each component inside the parentheses.
1. Simplify [tex]\(\frac{\sin \theta}{\sec \theta}\)[/tex]:
- Recall that [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex].
- So, [tex]\(\frac{\sin \theta}{\sec \theta} = \sin \theta \times \cos \theta\)[/tex].
Hence,
[tex]\[
\frac{\sin \theta}{\sec \theta} = \sin \theta \cos \theta
\][/tex]
2. Simplify [tex]\(\frac{\cos \theta}{\sin \theta}\)[/tex]:
- This is a straightforward simplification.
Hence,
[tex]\[
\frac{\cos \theta}{\sin \theta} = \frac{\cos \theta}{\sin \theta}
\][/tex]
### Step 2: Combine the simplified terms
Next, we rewrite the original expression with these simplified terms.
So the expression becomes:
[tex]\[
\frac{\sin \theta \cos \theta}{\frac{\cos \theta}{\sin \theta}}
\][/tex]
### Step 3: Simplify the combined fraction
To simplify this, we multiply by the reciprocal of the denominator.
[tex]\[
\frac{\sin \theta \cos \theta}{\frac{\cos \theta}{\sin \theta}} = \sin \theta \cos \theta \times \frac{\sin \theta}{\cos \theta}
\][/tex]
### Step 4: Simplify the multiplication
Now, we can cancel out the common terms [tex]\(\cos \theta\)[/tex]:
[tex]\[
\left(\sin \theta \cos \theta\right) \times \left(\frac{\sin \theta}{\cos \theta}\right) = \sin^2 \theta
\][/tex]
Finally, the simplified form of the given expression is:
[tex]\[
\sin^2 \theta
\][/tex]
Hence, our final answer is: [tex]\(\sin^2 \theta\)[/tex].
