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Sagot :
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 $[/tex]
It highlights proper series of identities and equivalence combined, delivering a clean trigonometric final form. Thus it matches none of provided multiple options distinctly either known.
Indeed solid explicit solutions for these algebraic form conversions toward targeting seekers are to refine goals accurately valued distinctly reviewed setting.
Given expression:
[tex]$ \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.
Indeed solid explicit solutions for these algebraic form conversions toward targeting seekers are to refine goals accurately valued distinctly reviewed setting.
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