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Sagot :
To determine the first acceptable step in simplifying the expression [tex]\(\frac{\cot x}{1-\csc x}\)[/tex], let's carefully consider how each given option might transform the expression.
A. [tex]\(\cot x - \cos x\)[/tex]:
This option does not seem to be a valid step because it suggests subtracting [tex]\(\cos x\)[/tex] from [tex]\(\cot x\)[/tex], which is not related to any known identities in simplifying the given expression. Therefore, this is not an acceptable first step.
B. [tex]\(\frac{\cot x(1+\csc x)}{(1-\csc x)(1+\csc x)}\)[/tex]:
This option suggests multiplying both the numerator and the denominator by [tex]\(1 + \csc x\)[/tex]. This technique uses the idea of rationalizing the denominator, which involves creating a difference of squares in the denominator:
[tex]\[ \frac{\cot x}{1 - \csc x} \cdot \frac{1 + \csc x}{1 + \csc x} = \frac{\cot x(1 + \csc x)}{(1 - \csc x)(1 + \csc x)} = \frac{\cot x(1 + \csc x)}{1 - (\csc x)^2} \][/tex]
We know from the Pythagorean identity that:
[tex]\[ \csc^2 x - 1 = \cot^2 x \quad \text{which implies} \quad 1 - \csc^2 x = -\cot^2 x \][/tex]
So, this results in:
[tex]\[ \frac{\cot x(1 + \csc x)}{-\cot^2 x} \][/tex]
Hence, Option B transforms the expression correctly through a valid identity and thus represents a correct first step.
C. Cannot be further simplified:
Given the valid transformations shown in Option B, the expression can indeed be simplified further. Thus, this option is incorrect.
D. [tex]\(\frac{\cot x(1-\csc x)}{(1-\csc x)(1+\csc x)}\)[/tex]:
This option suggests multiplying both the numerator and the denominator by [tex]\(1-\csc x\)[/tex], but this would not rationalize the denominator and would rather leave the denominator squared. This step does not lead us towards simplification:
[tex]\[ \frac{\cot x(1 - \csc x)}{(1 - \csc x)(1 + \csc x)} = \frac{\cot x(1 - \csc x)}{1 - (\csc x)^2} \][/tex]
This transformation does not utilize a rationalizing technique properly for simplification purposes. Therefore, this is not a valid first step.
Based on these analyses, the acceptable first step in simplifying the expression [tex]\(\frac{\cot x}{1-\csc x}\)[/tex] is provided by option B:
[tex]\[ \frac{\cot x(1+\csc x)}{(1-\csc x)(1+\csc x)} \][/tex]
A. [tex]\(\cot x - \cos x\)[/tex]:
This option does not seem to be a valid step because it suggests subtracting [tex]\(\cos x\)[/tex] from [tex]\(\cot x\)[/tex], which is not related to any known identities in simplifying the given expression. Therefore, this is not an acceptable first step.
B. [tex]\(\frac{\cot x(1+\csc x)}{(1-\csc x)(1+\csc x)}\)[/tex]:
This option suggests multiplying both the numerator and the denominator by [tex]\(1 + \csc x\)[/tex]. This technique uses the idea of rationalizing the denominator, which involves creating a difference of squares in the denominator:
[tex]\[ \frac{\cot x}{1 - \csc x} \cdot \frac{1 + \csc x}{1 + \csc x} = \frac{\cot x(1 + \csc x)}{(1 - \csc x)(1 + \csc x)} = \frac{\cot x(1 + \csc x)}{1 - (\csc x)^2} \][/tex]
We know from the Pythagorean identity that:
[tex]\[ \csc^2 x - 1 = \cot^2 x \quad \text{which implies} \quad 1 - \csc^2 x = -\cot^2 x \][/tex]
So, this results in:
[tex]\[ \frac{\cot x(1 + \csc x)}{-\cot^2 x} \][/tex]
Hence, Option B transforms the expression correctly through a valid identity and thus represents a correct first step.
C. Cannot be further simplified:
Given the valid transformations shown in Option B, the expression can indeed be simplified further. Thus, this option is incorrect.
D. [tex]\(\frac{\cot x(1-\csc x)}{(1-\csc x)(1+\csc x)}\)[/tex]:
This option suggests multiplying both the numerator and the denominator by [tex]\(1-\csc x\)[/tex], but this would not rationalize the denominator and would rather leave the denominator squared. This step does not lead us towards simplification:
[tex]\[ \frac{\cot x(1 - \csc x)}{(1 - \csc x)(1 + \csc x)} = \frac{\cot x(1 - \csc x)}{1 - (\csc x)^2} \][/tex]
This transformation does not utilize a rationalizing technique properly for simplification purposes. Therefore, this is not a valid first step.
Based on these analyses, the acceptable first step in simplifying the expression [tex]\(\frac{\cot x}{1-\csc x}\)[/tex] is provided by option B:
[tex]\[ \frac{\cot x(1+\csc x)}{(1-\csc x)(1+\csc x)} \][/tex]
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