Álgebra Exemplos

$\frac{\cot^{2} (x) \cos^{2} (x)}{\cot^{2} (x) - \cos^{2} (x)}$

Etapa 1

Simplifique o numerador.

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Etapa 1.1

Reescreva $\cot (x)$ em termos de senos e cossenos.

$\frac{{(\frac{\cos (x)}{\sin (x)})}^{2} \cos^{2} (x)}{\cot^{2} (x) - \cos^{2} (x)}$

Etapa 1.2

Aplique a regra do produto a $\frac{\cos (x)}{\sin (x)}$ .

$\frac{\frac{\cos^{2} (x)}{\sin^{2} (x)} \cos^{2} (x)}{\cot^{2} (x) - \cos^{2} (x)}$

Etapa 2

Simplifique o denominador.

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Etapa 2.1

Como os dois termos são quadrados perfeitos, fatore usando a fórmula da diferença de quadrados, $a^{2} - b^{2} = (a + b) (a - b)$ em que $a = \cot (x)$ e $b = \cos (x)$ .

$\frac{\frac{\cos^{2} (x)}{\sin^{2} (x)} \cos^{2} (x)}{(\cot (x) + \cos (x)) (\cot (x) - \cos (x))}$

Etapa 2.2

Simplifique.

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Etapa 2.2.1

Reescreva $\cot (x)$ em termos de senos e cossenos.

$\frac{\frac{\cos^{2} (x)}{\sin^{2} (x)} \cos^{2} (x)}{(\frac{\cos (x)}{\sin (x)} + \cos (x)) (\cot (x) - \cos (x))}$

Etapa 2.2.2

Fatore $\cos (x)$ de $\frac{\cos (x)}{\sin (x)} + \cos (x)$ .

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Etapa 2.2.2.1

Fatore $\cos (x)$ de $\frac{\cos (x)}{\sin (x)}$ .

$\frac{\frac{\cos^{2} (x)}{\sin^{2} (x)} \cos^{2} (x)}{(\cos (x) \frac{1}{\sin (x)} + \cos (x)) (\cot (x) - \cos (x))}$

Etapa 2.2.2.2

Multiplique por $1$ .

$\frac{\frac{\cos^{2} (x)}{\sin^{2} (x)} \cos^{2} (x)}{(\cos (x) \frac{1}{\sin (x)} + \cos (x) \cdot 1) (\cot (x) - \cos (x))}$

Etapa 2.2.2.3

Fatore $\cos (x)$ de $\cos (x) \frac{1}{\sin (x)} + \cos (x) \cdot 1$ .

$\frac{\frac{\cos^{2} (x)}{\sin^{2} (x)} \cos^{2} (x)}{\cos (x) (\frac{1}{\sin (x)} + 1) (\cot (x) - \cos (x))}$

Etapa 2.2.3

Reescreva $\cot (x)$ em termos de senos e cossenos.

$\frac{\frac{\cos^{2} (x)}{\sin^{2} (x)} \cos^{2} (x)}{\cos (x) (\frac{1}{\sin (x)} + 1) (\frac{\cos (x)}{\sin (x)} - \cos (x))}$

Etapa 2.2.4

Fatore $\cos (x)$ de $\frac{\cos (x)}{\sin (x)} - \cos (x)$ .

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Etapa 2.2.4.1

Fatore $\cos (x)$ de $\frac{\cos (x)}{\sin (x)}$ .

$\frac{\frac{\cos^{2} (x)}{\sin^{2} (x)} \cos^{2} (x)}{\cos (x) (\frac{1}{\sin (x)} + 1) (\cos (x) \frac{1}{\sin (x)} - \cos (x))}$

Etapa 2.2.4.2

Fatore $\cos (x)$ de $- \cos (x)$ .

$\frac{\frac{\cos^{2} (x)}{\sin^{2} (x)} \cos^{2} (x)}{\cos (x) (\frac{1}{\sin (x)} + 1) (\cos (x) \frac{1}{\sin (x)} + \cos (x) \cdot - 1)}$

Etapa 2.2.4.3

Fatore $\cos (x)$ de $\cos (x) \frac{1}{\sin (x)} + \cos (x) \cdot - 1$ .

$\frac{\frac{\cos^{2} (x)}{\sin^{2} (x)} \cos^{2} (x)}{\cos (x) (\frac{1}{\sin (x)} + 1) (\cos (x) (\frac{1}{\sin (x)} - 1))}$

$\frac{\frac{\cos^{2} (x)}{\sin^{2} (x)} \cos^{2} (x)}{\cos (x) (\frac{1}{\sin (x)} + 1) \cos (x) (\frac{1}{\sin (x)} - 1)}$

Etapa 2.2.5

Combine expoentes.

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Etapa 2.2.5.1

Eleve $\cos (x)$ à potência de $1$ .

$\frac{\frac{\cos^{2} (x)}{\sin^{2} (x)} \cos^{2} (x)}{\cos^{1} (x) \cos (x) (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1)}$

Etapa 2.2.5.2

Eleve $\cos (x)$ à potência de $1$ .

$\frac{\frac{\cos^{2} (x)}{\sin^{2} (x)} \cos^{2} (x)}{\cos^{1} (x) \cos^{1} (x) (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1)}$

Etapa 2.2.5.3

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{\frac{\cos^{2} (x)}{\sin^{2} (x)} \cos^{2} (x)}{{\cos (x)}^{1 + 1} (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1)}$

Etapa 2.2.5.4

Some $1$ e $1$ .

$\frac{\frac{\cos^{2} (x)}{\sin^{2} (x)} \cos^{2} (x)}{\cos^{2} (x) (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1)}$

Etapa 3

Combine $\frac{\cos^{2} (x)}{\sin^{2} (x)}$ e $\cos^{2} (x)$ .

$\frac{\frac{\cos^{2} (x) \cos^{2} (x)}{\sin^{2} (x)}}{\cos^{2} (x) (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1)}$

Etapa 4

Multiplique $\cos^{2} (x)$ por $\cos^{2} (x)$ somando os expoentes.

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Etapa 4.1

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$\frac{\frac{{\cos (x)}^{2 + 2}}{\sin^{2} (x)}}{\cos^{2} (x) (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1)}$

Etapa 4.2

Some $2$ e $2$ .

$\frac{\frac{\cos^{4} (x)}{\sin^{2} (x)}}{\cos^{2} (x) (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1)}$

Etapa 5

Multiplique o numerador pelo inverso do denominador.

$\frac{\cos^{4} (x)}{\sin^{2} (x)} \cdot \frac{1}{\cos^{2} (x) (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1)}$

Etapa 6

Combine.

$\frac{\cos^{4} (x) \cdot 1}{\sin^{2} (x) (\cos^{2} (x) (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1))}$

Etapa 7

Cancele o fator comum de $\cos^{4} (x)$ e $\cos^{2} (x)$ .

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Etapa 7.1

Fatore $\cos^{2} (x)$ de $\cos^{4} (x) \cdot 1$ .

$\frac{\cos^{2} (x) (\cos^{2} (x) \cdot 1)}{\sin^{2} (x) (\cos^{2} (x) (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1))}$

Etapa 7.2

Cancele os fatores comuns.

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Etapa 7.2.1

Fatore $\cos^{2} (x)$ de $\sin^{2} (x) (\cos^{2} (x) (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1))$ .

$\frac{\cos^{2} (x) (\cos^{2} (x) \cdot 1)}{\cos^{2} (x) (\sin^{2} (x) ((\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1)))}$

Etapa 7.2.2

Cancele o fator comum.

$\frac{\cos^{2} (x) (\cos^{2} (x) \cdot 1)}{\cos^{2} (x) (\sin^{2} (x) ((\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1)))}$

Etapa 7.2.3

Reescreva a expressão.

$\frac{\cos^{2} (x) \cdot 1}{\sin^{2} (x) ((\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1))}$

Etapa 8

Multiplique $\cos^{2} (x)$ por $1$ .

$\frac{\cos^{2} (x)}{\sin^{2} (x) (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1)}$

Etapa 9

Multiplique por $1$ .

$\frac{\cos^{2} (x) \cdot 1}{\sin^{2} (x) (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1)}$

Etapa 10

Multiplique por $1$ .

$\frac{\cos^{2} (x) \cdot 1}{\sin^{2} (x) \cdot 1 (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1)}$

Etapa 11

Separe as frações.

$\frac{1}{(1 (\frac{1}{\sin (x)} + 1)) (\frac{1}{\sin (x)} - 1)} \cdot \frac{\cos^{2} (x)}{\sin^{2} (x)}$

Etapa 12

Converta de $\frac{\cos^{2} (x)}{\sin^{2} (x)}$ em $\cot^{2} (x)$ .

$\frac{1}{(1 (\frac{1}{\sin (x)} + 1)) (\frac{1}{\sin (x)} - 1)} \cot^{2} (x)$

Etapa 13

Multiplique $\frac{1}{\sin (x)} + 1$ por $1$ .

$\frac{1}{(\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1)} \cot^{2} (x)$

Etapa 14

Simplifique o denominador.

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Etapa 14.1

Converta de $\frac{1}{\sin (x)}$ em $\csc (x)$ .

$\frac{1}{(\csc (x) + 1) (\frac{1}{\sin (x)} - 1)} \cot^{2} (x)$

Etapa 14.2

Converta de $\frac{1}{\sin (x)}$ em $\csc (x)$ .

$\frac{1}{(\csc (x) + 1) (\csc (x) - 1)} \cot^{2} (x)$

Etapa 15

Combine $\frac{1}{(\csc (x) + 1) (\csc (x) - 1)}$ e $\cot^{2} (x)$ .

$\frac{\cot^{2} (x)}{(\csc (x) + 1) (\csc (x) - 1)}$