To simplify the given trigonometric expression, let's break it down step by step:
Given expression:
[tex]$ \frac{\cot (\theta) \cos (\theta)}{\sin (\theta)} \cdot \tan (\theta) + \frac{\sin (\theta)}{\cos (\theta) \tan (\theta)} $[/tex]
### Step 1: Handling the first part
[tex]$ \frac{\cot (\theta) \cos (\theta)}{\sin (\theta)} \cdot \tan (\theta) $[/tex]
Recall the trigonometric identities:
[tex]$ \cot (\theta) = \frac{\cos (\theta)}{\sin (\theta)} \quad \text{and} \quad \tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)} $[/tex]
First, substitute [tex]\( \cot (\theta) \)[/tex]:
[tex]$ \frac{\left( \frac{\cos (\theta)}{\sin (\theta)} \right) \cos (\theta)}{\sin (\theta)} \cdot \tan (\theta) = \frac{\cos^2 (\theta)}{\sin^2 (\theta)} \cdot \tan (\theta) $[/tex]
Now, substitute [tex]\( \tan (\theta) \)[/tex]:
[tex]$ \frac{\cos^2(\theta)}{\sin^2(\theta)} \cdot \frac{\sin(\theta)}{\cos(\theta)} $[/tex]
Simplify the expression step-by-step:
[tex]$ \frac{\cos^2 (\theta)}{\sin^2 (\theta)} \cdot \frac{\sin (\theta)}{\cos (\theta)} = \frac{\cos (\theta) \cdot \cos (\theta) \cdot \sin (\theta)}{\sin^2(\theta) \cdot \cos (\theta)} = \frac{\cos (\theta) \cos (\theta)}{\sin (\theta) \sin (\theta)} \cdot \frac{\sin (\theta)}{\cos (\theta)} $[/tex]
[tex]$ \frac{\cos (\theta) \cos (\theta)}{\sin (\theta) \sin (\theta)} \cdot \frac{\sin (\theta)}{\cos (\theta)} = \frac{\cos (\theta)}{\sin (\theta)} = \cot (\theta) $[/tex]
So, the first part simplifies to:
[tex]$ \cot (\theta) $[/tex]
### Step 2: Handling the second part
[tex]$ \frac{\sin (\theta)}{\cos (\theta) \tan (\theta)} $[/tex]
Again, use the identity for [tex]\( \tan (\theta) \)[/tex]:
[tex]$ \tan (\theta) = \frac{\sin (\theta)}{\cos (\theta)} $[/tex]
Substitute [tex]\( \tan (\theta) \)[/tex]:
[tex]$ \frac{\sin (\theta)}{\cos (\theta) \cdot \frac{\sin (\theta)}{\cos (\theta)}} = \frac{\sin (\theta)}{\frac{\sin (\theta) \cdot \cos (\theta)}{\cos (\theta)}} = \frac{\sin (\theta)}{\sin (\theta)} = 1 $[/tex]
### Step 3: Combine both simplified parts
[tex]$ \cot (\theta) + 1 $[/tex]
However, rechecking the options, none match [tex]\( \cot (\theta) + 1 \)[/tex], which could indicate a mistake in simplification. After re-evaluating, the combined and simplified expression/trigonometrical identities confirms consistently:
The original simplified form:
[tex]$ \cot (\theta) + 1 $[/tex]
It highlights proper series of identities and equivalence combined, delivering a clean trigonometric final form. Thus it matches none of provided multiple options distinctly either known.
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