Let's simplify the given expression step by step. The expression to simplify is:
[tex]\[
\frac{\cot (\theta) \cos (\theta)}{\sin (\theta)} \cdot \tan (\theta) \div \frac{\sin (\theta)}{\cos (\theta) \tan (\theta)}
\][/tex]
First, we rewrite [tex]\(\div\)[/tex] as the multiplication by the reciprocal:
[tex]\[
\frac{\cot (\theta) \cos (\theta)}{\sin (\theta)} \cdot \tan (\theta) \cdot \frac{\cos (\theta) \tan (\theta)}{\sin (\theta)}
\][/tex]
Now, let's simplify each component of the expression:
1. [tex]\(\cot(\theta)\)[/tex] is defined as [tex]\(\frac{\cos(\theta)}{\sin(\theta)}\)[/tex].
So,
[tex]\[
\frac{\cot (\theta) \cos (\theta)}{\sin (\theta)} = \frac{\left( \frac{\cos (\theta)}{\sin (\theta)} \right) \cos (\theta)}{\sin (\theta)} = \frac{\cos^2 (\theta)}{\sin^2 (\theta)}
\][/tex]
2. The expression now becomes:
[tex]\[
\frac{\cos^2 (\theta)}{\sin^2 (\theta)} \cdot \tan (\theta) \cdot \frac{\cos (\theta) \tan (\theta)}{\sin (\theta)}
\][/tex]
Recall that [tex]\(\tan(\theta)\)[/tex] is defined as [tex]\(\frac{\sin(\theta)}{\cos(\theta)}\)[/tex]. Substitute [tex]\(\tan(\theta)\)[/tex] into the expression:
[tex]\[
\frac{\cos^2 (\theta)}{\sin^2 (\theta)} \cdot \frac{\sin(\theta)}{\cos(\theta)} \cdot \frac{\cos (\theta) \frac{\sin(\theta)}{\cos(\theta)}}{\sin (\theta)}
\][/tex]
We can simplify this step-by-step:
3. Simplify [tex]\(\frac{\sin(\theta)}{\cos(\theta)}\)[/tex]:
[tex]\[
\frac{\cos^2 (\theta)}{\sin^2 (\theta)} \cdot \frac{\sin (\theta)}{\cos (\theta)}
\][/tex]
4. Notice that [tex]\(\frac{\cos (\theta) \frac{\sin(\theta)}{\cos (\theta)}}{\sin (\theta)} = \frac{\sin(\theta)}{\sin (\theta)}\)[/tex] simplifies to 1:
5. Now the expression reduces to:
[tex]\[
\frac{\cos^2 (\theta)}{\sin^2 (\theta)} \cdot \frac{\sin (\theta)}{\cos (\theta)} \cdot 1
\][/tex]
6. Let's combine the simplified components:
[tex]\[
\frac{\cos^2 (\theta)}{\sin^2 (\theta)} \cdot \frac{\sin (\theta)}{\cos (\theta)} = \frac{\cos (\theta)}{\sin^2 (\theta)} \cdot \sin (\theta) = \frac{\cos (\theta)\sin (\theta)}{\sin^2 (\theta)}
\][/tex]
7. Now, simplify [tex]\(\frac{\cos (\theta)}{\sin (\theta)}\)[/tex]:
[tex]\[
\frac{\cos (\theta)}{\sin (\theta)} = \cot (\theta)
\][/tex]
Thus, the simplest form of the given expression simplifies to:
[tex]\[
\cot (\theta)
\][/tex]
So, the correct answer is:
[tex]\(\boxed{\cot (\theta)}\)[/tex]
