Trigonometria Exemplos

$\frac{\tan (x) + \sec (x) - 1}{\tan (x) - \sec (x) + 1} = \tan (x) + \sec (x)$

Etapa 1

Simplifique o lado esquerdo.

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Etapa 1.1

Simplifique $\frac{\tan (x) + \sec (x) - 1}{\tan (x) - \sec (x) + 1}$ .

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Etapa 1.1.1

Simplifique o numerador.

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Etapa 1.1.1.1

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

$\frac{\frac{\sin (x)}{\cos (x)} + \sec (x) - 1}{\tan (x) - \sec (x) + 1} = \tan (x) + \sec (x)$

Etapa 1.1.1.2

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

$\frac{\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1}{\tan (x) - \sec (x) + 1} = \tan (x) + \sec (x)$

Etapa 1.1.2

Simplifique o denominador.

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Etapa 1.1.2.1

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

$\frac{\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1}{\frac{\sin (x)}{\cos (x)} - \sec (x) + 1} = \tan (x) + \sec (x)$

Etapa 1.1.2.2

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

$\frac{\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1}{\frac{\sin (x)}{\cos (x)} - \frac{1}{\cos (x)} + 1} = \tan (x) + \sec (x)$

Etapa 1.1.3

Multiply the numerator and denominator of the fraction by $\cos (x)$ .

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Etapa 1.1.3.1

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

$\frac{\cos (x)}{\cos (x)} \cdot \frac{\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1}{\frac{\sin (x)}{\cos (x)} - \frac{1}{\cos (x)} + 1} = \tan (x) + \sec (x)$

Etapa 1.1.3.2

Combine.

$\frac{\cos (x) (\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1)}{\cos (x) (\frac{\sin (x)}{\cos (x)} - \frac{1}{\cos (x)} + 1)} = \tan (x) + \sec (x)$

Etapa 1.1.4

Aplique a propriedade distributiva.

$\frac{\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1}{\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{1}{\cos (x)}) + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Etapa 1.1.5

Simplifique cancelando.

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Etapa 1.1.5.1

Cancele o fator comum de $\cos (x)$ .

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Etapa 1.1.5.1.1

Cancele o fator comum.

Etapa 1.1.5.1.2

Reescreva a expressão.

$\frac{\sin (x) + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1}{\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{1}{\cos (x)}) + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Etapa 1.1.5.2

Cancele o fator comum de $\cos (x)$ .

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Etapa 1.1.5.2.1

Cancele o fator comum.

$\frac{\sin (x) + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1}{\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{1}{\cos (x)}) + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Etapa 1.1.5.2.2

Reescreva a expressão.

$\frac{\sin (x) + 1 + \cos (x) \cdot - 1}{\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{1}{\cos (x)}) + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Etapa 1.1.5.3

Cancele o fator comum de $\cos (x)$ .

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Etapa 1.1.5.3.1

Cancele o fator comum.

$\frac{\sin (x) + 1 + \cos (x) \cdot - 1}{\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{1}{\cos (x)}) + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Etapa 1.1.5.3.2

Reescreva a expressão.

$\frac{\sin (x) + 1 + \cos (x) \cdot - 1}{\sin (x) + \cos (x) (- \frac{1}{\cos (x)}) + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Etapa 1.1.5.4

Cancele o fator comum de $\cos (x)$ .

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Etapa 1.1.5.4.1

Mova o negativo de maior ordem em $- \frac{1}{\cos (x)}$ para o numerador.

$\frac{\sin (x) + 1 + \cos (x) \cdot - 1}{\sin (x) + \cos (x) \frac{- 1}{\cos (x)} + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Etapa 1.1.5.4.2

Cancele o fator comum.

$\frac{\sin (x) + 1 + \cos (x) \cdot - 1}{\sin (x) + \cos (x) \frac{- 1}{\cos (x)} + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Etapa 1.1.5.4.3

Reescreva a expressão.

$\frac{\sin (x) + 1 + \cos (x) \cdot - 1}{\sin (x) - 1 + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Etapa 1.1.6

Simplifique o numerador.

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Etapa 1.1.6.1

Mova $- 1$ para a esquerda de $\cos (x)$ .

$\frac{\sin (x) + 1 - 1 \cdot \cos (x)}{\sin (x) - 1 + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Etapa 1.1.6.2

Reescreva $- 1 \cos (x)$ como $- \cos (x)$ .

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Etapa 1.1.7

Multiplique $\cos (x)$ por $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 2

Simplifique o lado direito.

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Etapa 2.1

Simplifique cada termo.

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Etapa 2.1.1

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

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \sec (x)$

Etapa 2.1.2

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

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

Etapa 3

Multiplique os dois lados da equação por $\cos (x)$ .

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

Etapa 4

Combine $\cos (x)$ e $\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)}$ .

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

Etapa 5

Aplique a propriedade distributiva.

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

Etapa 6

Cancele o fator comum de $\cos (x)$ .

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Etapa 6.1

Cancele o fator comum.

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

Etapa 6.2

Reescreva a expressão.

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

Etapa 7

Cancele o fator comum de $\cos (x)$ .

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Etapa 7.1

Cancele o fator comum.

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

Etapa 7.2

Reescreva a expressão.

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

Etapa 8

Divida cada termo na equação por $\cos (x)$ .

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

Etapa 9

Separe as frações.

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

Etapa 10

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\frac{\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}}{1} \cdot \sec (x) = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 11

Divida $\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ por $1$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} \cdot \sec (x) = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 12

Combine $\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ e $\sec (x)$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x)) \sec (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 13

Reordene os fatores em $\frac{\cos (x) (\sin (x) + 1 - \cos (x)) \sec (x)}{\sin (x) - 1 + \cos (x)}$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 14

Simplifique cada termo.

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Etapa 14.1

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \frac{1}{\cos (x)}$

Etapa 14.2

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 15

Simplifique o lado esquerdo.

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Etapa 15.1

Simplifique $\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ .

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Etapa 15.1.1

Simplifique o numerador.

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Etapa 15.1.1.1

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

$\frac{\cos (x) \frac{1}{\cos (x)} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 15.1.1.2

Combine $\cos (x)$ e $\frac{1}{\cos (x)}$ .

$\frac{\frac{\cos (x)}{\cos (x)} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 15.1.1.3

Reduza a expressão cancelando os fatores comuns.

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Etapa 15.1.1.3.1

Reduza a expressão $\frac{\cos (x)}{\cos (x)}$ cancelando os fatores comuns.

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Etapa 15.1.1.3.1.1

Cancele o fator comum.

$\frac{\frac{\cos (x)}{\cos (x)} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 15.1.1.3.1.2

Reescreva a expressão.

$\frac{\frac{1}{1} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 15.1.1.3.2

Reescreva a expressão.

$\frac{1 (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 15.1.2

Multiplique $\sin (x) + 1 - \cos (x)$ por $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 16

Simplifique o lado direito.

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Etapa 16.1

Simplifique cada termo.

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Etapa 16.1.1

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

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \sec (x)$

Etapa 16.1.2

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

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

Etapa 17

Multiplique os dois lados da equação por $\cos (x)$ .

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

Etapa 18

Combine $\cos (x)$ e $\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)}$ .

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

Etapa 19

Aplique a propriedade distributiva.

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

Etapa 20

Cancele o fator comum de $\cos (x)$ .

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Etapa 20.1

Cancele o fator comum.

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

Etapa 20.2

Reescreva a expressão.

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

Etapa 21

Cancele o fator comum de $\cos (x)$ .

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Etapa 21.1

Cancele o fator comum.

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

Etapa 21.2

Reescreva a expressão.

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

Etapa 22

Divida cada termo na equação por $\cos (x)$ .

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

Etapa 23

Separe as frações.

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

Etapa 24

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\frac{\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}}{1} \cdot \sec (x) = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 25

Divida $\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ por $1$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} \cdot \sec (x) = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 26

Combine $\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ e $\sec (x)$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x)) \sec (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 27

Reordene os fatores em $\frac{\cos (x) (\sin (x) + 1 - \cos (x)) \sec (x)}{\sin (x) - 1 + \cos (x)}$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 28

Simplifique cada termo.

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Etapa 28.1

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \frac{1}{\cos (x)}$

Etapa 28.2

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 29

Simplifique o lado esquerdo.

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Etapa 29.1

Simplifique $\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ .

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Etapa 29.1.1

Simplifique o numerador.

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Etapa 29.1.1.1

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

$\frac{\cos (x) \frac{1}{\cos (x)} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 29.1.1.2

Combine $\cos (x)$ e $\frac{1}{\cos (x)}$ .

$\frac{\frac{\cos (x)}{\cos (x)} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 29.1.1.3

Reduza a expressão cancelando os fatores comuns.

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Etapa 29.1.1.3.1

Reduza a expressão $\frac{\cos (x)}{\cos (x)}$ cancelando os fatores comuns.

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Etapa 29.1.1.3.1.1

Cancele o fator comum.

$\frac{\frac{\cos (x)}{\cos (x)} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 29.1.1.3.1.2

Reescreva a expressão.

$\frac{\frac{1}{1} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 29.1.1.3.2

Reescreva a expressão.

$\frac{1 (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 29.1.2

Multiplique $\sin (x) + 1 - \cos (x)$ por $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 30

Simplifique o lado direito.

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Etapa 30.1

Simplifique cada termo.

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Etapa 30.1.1

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

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \sec (x)$

Etapa 30.1.2

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

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

Etapa 31

Multiplique os dois lados da equação por $\cos (x)$ .

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

Etapa 32

Combine $\cos (x)$ e $\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)}$ .

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

Etapa 33

Aplique a propriedade distributiva.

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

Etapa 34

Cancele o fator comum de $\cos (x)$ .

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Etapa 34.1

Cancele o fator comum.

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

Etapa 34.2

Reescreva a expressão.

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

Etapa 35

Cancele o fator comum de $\cos (x)$ .

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Etapa 35.1

Cancele o fator comum.

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

Etapa 35.2

Reescreva a expressão.

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

Etapa 36

Divida cada termo na equação por $\cos (x)$ .

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

Etapa 37

Separe as frações.

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

Etapa 38

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\frac{\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}}{1} \cdot \sec (x) = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 39

Divida $\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ por $1$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} \cdot \sec (x) = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 40

Combine $\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ e $\sec (x)$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x)) \sec (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 41

Reordene os fatores em $\frac{\cos (x) (\sin (x) + 1 - \cos (x)) \sec (x)}{\sin (x) - 1 + \cos (x)}$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 42

Simplifique cada termo.

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Etapa 42.1

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \frac{1}{\cos (x)}$

Etapa 42.2

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 43

Simplifique o lado esquerdo.

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Etapa 43.1

Simplifique $\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ .

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Etapa 43.1.1

Simplifique o numerador.

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Etapa 43.1.1.1

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

$\frac{\cos (x) \frac{1}{\cos (x)} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 43.1.1.2

Combine $\cos (x)$ e $\frac{1}{\cos (x)}$ .

$\frac{\frac{\cos (x)}{\cos (x)} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 43.1.1.3

Reduza a expressão cancelando os fatores comuns.

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Etapa 43.1.1.3.1

Reduza a expressão $\frac{\cos (x)}{\cos (x)}$ cancelando os fatores comuns.

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Etapa 43.1.1.3.1.1

Cancele o fator comum.

$\frac{\frac{\cos (x)}{\cos (x)} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 43.1.1.3.1.2

Reescreva a expressão.

$\frac{\frac{1}{1} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 43.1.1.3.2

Reescreva a expressão.

$\frac{1 (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 43.1.2

Multiplique $\sin (x) + 1 - \cos (x)$ por $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 44

Simplifique o lado direito.

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Etapa 44.1

Simplifique cada termo.

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Etapa 44.1.1

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

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \sec (x)$

Etapa 44.1.2

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

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

Etapa 45

Multiplique os dois lados da equação por $\cos (x)$ .

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

Etapa 46

Combine $\cos (x)$ e $\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)}$ .

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

Etapa 47

Aplique a propriedade distributiva.

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

Etapa 48

Cancele o fator comum de $\cos (x)$ .

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Etapa 48.1

Cancele o fator comum.

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

Etapa 48.2

Reescreva a expressão.

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

Etapa 49

Cancele o fator comum de $\cos (x)$ .

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Etapa 49.1

Cancele o fator comum.

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

Etapa 49.2

Reescreva a expressão.

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

Etapa 50

Divida cada termo na equação por $\cos (x)$ .

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

Etapa 51

Separe as frações.

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

Etapa 52

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\frac{\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}}{1} \cdot \sec (x) = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 53

Divida $\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ por $1$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} \cdot \sec (x) = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 54

Combine $\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ e $\sec (x)$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x)) \sec (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 55

Reordene os fatores em $\frac{\cos (x) (\sin (x) + 1 - \cos (x)) \sec (x)}{\sin (x) - 1 + \cos (x)}$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 56

Simplifique cada termo.

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Etapa 56.1

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \frac{1}{\cos (x)}$

Etapa 56.2

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Etapa 57

Divida cada termo na equação por $\cos (x)$ .

$\frac{\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}}{\cos (x)} = \frac{\tan (x)}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Etapa 58

Multiplique o numerador pelo inverso do denominador.

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} \cdot \frac{1}{\cos (x)} = \frac{\tan (x)}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Etapa 59

Cancele o fator comum de $\cos (x)$ .

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Etapa 59.1

Fatore $\cos (x)$ de $\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))$ .

$\frac{\cos (x) (\sec (x) (\sin (x) + 1 - \cos (x)))}{\sin (x) - 1 + \cos (x)} \cdot \frac{1}{\cos (x)} = \frac{\tan (x)}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Etapa 59.2

Cancele o fator comum.

$\frac{\cos (x) (\sec (x) (\sin (x) + 1 - \cos (x)))}{\sin (x) - 1 + \cos (x)} \cdot \frac{1}{\cos (x)} = \frac{\tan (x)}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Etapa 59.3

Reescreva a expressão.

$\frac{\sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\tan (x)}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Etapa 60

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

$\frac{\frac{1}{\cos (x)} \cdot (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\tan (x)}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Etapa 61

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

$\frac{1}{\cos (x)} \cdot \frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \frac{\tan (x)}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Etapa 62

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

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\tan (x)}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Etapa 63

Simplifique cada termo.

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Etapa 63.1

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

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Etapa 63.2

Reescreva $\frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)}$ como um produto.

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Etapa 63.3

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

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\sin (x)}{\cos (x) \cos (x)} + \frac{\sec (x)}{\cos (x)}$

Etapa 63.4

Simplifique o denominador.

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Etapa 63.4.1

Eleve $\cos (x)$ à potência de $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\sin (x)}{\cos (x) \cos (x)} + \frac{\sec (x)}{\cos (x)}$

Etapa 63.4.2

Eleve $\cos (x)$ à potência de $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\sin (x)}{\cos (x) \cos (x)} + \frac{\sec (x)}{\cos (x)}$

Etapa 63.4.3

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

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

Etapa 63.4.4

Some $1$ e $1$ .

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

Etapa 63.5

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

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

Etapa 63.6

Reescreva $\frac{\frac{1}{\cos (x)}}{\cos (x)}$ como um produto.

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

Etapa 63.7

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

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

Etapa 63.8

Simplifique o denominador.

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Etapa 63.8.1

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 63.8.2

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 63.8.3

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

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

Etapa 63.8.4

Some $1$ e $1$ .

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

Etapa 64

Simplifique cada termo.

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Etapa 64.1

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

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

Etapa 64.2

Separe as frações.

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

Etapa 64.3

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

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

Etapa 64.4

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

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

Etapa 64.5

Reescreva $1$ como $1^{2}$ .

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

Etapa 64.6

Reescreva $\frac{1^{2}}{\cos^{2} (x)}$ como ${(\frac{1}{\cos (x)})}^{2}$ .

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

Etapa 64.7

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

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

Etapa 65

Divida cada termo na equação por $\cos (x)$ .

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

Etapa 66

Separe as frações.

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

Etapa 67

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

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

Etapa 68

Divida $\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))}$ por $1$ .

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

Etapa 69

Combine $\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))}$ e $\sec (x)$ .

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

Etapa 70

Separe as frações.

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

Etapa 71

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

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

Etapa 72

Reescreva $\frac{\frac{1}{\cos (x)}}{\cos (x)}$ como um produto.

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

Etapa 73

Simplifique.

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Etapa 73.1

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

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

Etapa 73.2

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

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

Etapa 73.3

Eleve $\sec (x)$ à potência de $1$ .

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

Etapa 73.4

Eleve $\sec (x)$ à potência de $1$ .

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

Etapa 73.5

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

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

Etapa 73.6

Some $1$ e $1$ .

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

Etapa 74

Combine $\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)}$ e $\sec^{2} (x)$ .

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

Etapa 75

Reordene os fatores em $\frac{(\sin (x) + 1 - \cos (x)) \sec^{2} (x)}{\sin (x) - 1 + \cos (x)}$ .

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

Etapa 76

Simplifique cada termo.

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Etapa 76.1

Separe as frações.

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

Etapa 76.2

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

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

Etapa 76.3

Reescreva $\frac{\frac{1}{\cos (x)}}{\cos (x)}$ como um produto.

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

Etapa 76.4

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

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

Etapa 76.5

Divida $\tan (x)$ por $1$ .

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

Etapa 76.6

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

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

Etapa 76.7

Simplifique o denominador.

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Etapa 76.7.1

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 76.7.2

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 76.7.3

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

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

Etapa 76.7.4

Some $1$ e $1$ .

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

Etapa 76.8

Combine.

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

Etapa 76.9

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

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Etapa 76.9.1

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

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Etapa 76.9.1.1

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 76.9.1.2

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

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

Etapa 76.9.2

Some $1$ e $2$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x) \cdot 1}{\cos^{3} (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Etapa 76.10

Multiplique $\sin (x)$ por $1$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Etapa 76.11

Simplifique o numerador.

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Etapa 76.11.1

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

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{{(\frac{1}{\cos (x)})}^{2}}{\cos (x)}$

Etapa 76.11.2

Aplique a regra do produto a $\frac{1}{\cos (x)}$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{\frac{1^{2}}{\cos^{2} (x)}}{\cos (x)}$

Etapa 76.11.3

Um elevado a qualquer potência é um.

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{\frac{1}{\cos^{2} (x)}}{\cos (x)}$

Etapa 76.12

Multiplique o numerador pelo inverso do denominador.

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{1}{\cos^{2} (x)} \cdot \frac{1}{\cos (x)}$

Etapa 76.13

Combine.

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{1 \cdot 1}{\cos^{2} (x) \cos (x)}$

Etapa 76.14

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

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Etapa 76.14.1

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

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Etapa 76.14.1.1

Eleve $\cos (x)$ à potência de $1$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{1 \cdot 1}{\cos^{2} (x) \cos (x)}$

Etapa 76.14.1.2

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

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{1 \cdot 1}{{\cos (x)}^{2 + 1}}$

Etapa 76.14.2

Some $2$ e $1$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{1 \cdot 1}{\cos^{3} (x)}$

Etapa 76.15

Multiplique $1$ por $1$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{1}{\cos^{3} (x)}$

Etapa 77

Simplifique cada termo.

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Etapa 77.1

Fatore $\cos (x)$ de $\cos^{3} (x)$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x) \cos^{2} (x)} + \frac{1}{\cos^{3} (x)}$

Etapa 77.2

Separe as frações.

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{1}{\cos^{2} (x)} \cdot \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos^{3} (x)}$

Etapa 77.3

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{1}{\cos^{2} (x)} \cdot \tan (x) + \frac{1}{\cos^{3} (x)}$

Etapa 77.4

Reescreva $1$ como $1^{2}$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{1^{2}}{\cos^{2} (x)} \cdot \tan (x) + \frac{1}{\cos^{3} (x)}$

Etapa 77.5

Reescreva $\frac{1^{2}}{\cos^{2} (x)}$ como ${(\frac{1}{\cos (x)})}^{2}$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = {(\frac{1}{\cos (x)})}^{2} \tan (x) + \frac{1}{\cos^{3} (x)}$

Etapa 77.6

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}$

Etapa 78

Substitua $\sec^{2} (x)$ por $1 + \tan^{2} (x)$ com base na identidade $\tan^{2} (x) + 1 = \sec^{2} (x)$ .

$\frac{(1 + \tan^{2} (x)) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}$

Etapa 79

Multiplique os dois lados por $\sin (x) - 1 + \cos (x)$ .

$\frac{(1 + \tan^{2} (x)) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} (\sin (x) - 1 + \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80

Simplifique.

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Etapa 80.1

Simplifique o lado esquerdo.

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Etapa 80.1.1

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

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Etapa 80.1.1.1

Cancele o fator comum de $\sin (x) - 1 + \cos (x)$ .

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Etapa 80.1.1.1.1

Cancele o fator comum.

$\frac{(1 + \tan^{2} (x)) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} (\sin (x) - 1 + \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.1.2

Reescreva a expressão.

$(1 + \tan^{2} (x)) (\sin (x) + 1 - \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.2

Reorganize os termos.

$(\tan^{2} (x) + 1) (\sin (x) + 1 - \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.3

Aplique a identidade trigonométrica fundamental.

$\sec^{2} (x) (\sin (x) + 1 - \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.4

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

${(\frac{1}{\cos (x)})}^{2} (\sin (x) + 1 - \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.5

Simplifique a expressão.

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Etapa 80.1.1.5.1

Aplique a regra do produto a $\frac{1}{\cos (x)}$ .

$\frac{1^{2}}{\cos^{2} (x)} (\sin (x) + 1 - \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.5.2

Um elevado a qualquer potência é um.

$\frac{1}{\cos^{2} (x)} (\sin (x) + 1 - \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.6

Aplique a propriedade distributiva.

$\frac{1}{\cos^{2} (x)} \sin (x) + \frac{1}{\cos^{2} (x)} \cdot 1 + \frac{1}{\cos^{2} (x)} (- \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.7

Simplifique.

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Etapa 80.1.1.7.1

Combine $\frac{1}{\cos^{2} (x)}$ e $\sin (x)$ .

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} \cdot 1 + \frac{1}{\cos^{2} (x)} (- \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.7.2

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

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} (- \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.7.3

Reescreva usando a propriedade comutativa da multiplicação.

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos^{2} (x)} \cos (x) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.8

Simplifique cada termo.

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Etapa 80.1.1.8.1

Cancele o fator comum de $\cos (x)$ .

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Etapa 80.1.1.8.1.1

Mova o negativo de maior ordem em $- \frac{1}{\cos^{2} (x)}$ para o numerador.

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} + \frac{- 1}{\cos^{2} (x)} \cos (x) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.8.1.2

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

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} + \frac{- 1}{\cos (x) \cos (x)} \cos (x) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.8.1.3

Cancele o fator comum.

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} + \frac{- 1}{\cos (x) \cos (x)} \cos (x) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.8.1.4

Reescreva a expressão.

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} + \frac{- 1}{\cos (x)} = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.8.2

Mova o número negativo para a frente da fração.

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.9

Simplifique cada termo.

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Etapa 80.1.1.9.1

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

$\frac{\sin (x)}{\cos (x) \cos (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.9.2

Separe as frações.

$\frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.9.3

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

$\frac{1}{\cos (x)} \tan (x) + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.9.4

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\sec (x) \tan (x) + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.9.5

Reescreva $1$ como $1^{2}$ .

$\sec (x) \tan (x) + \frac{1^{2}}{\cos^{2} (x)} - \frac{1}{\cos (x)} = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.9.6

Reescreva $\frac{1^{2}}{\cos^{2} (x)}$ como ${(\frac{1}{\cos (x)})}^{2}$ .

$\sec (x) \tan (x) + {(\frac{1}{\cos (x)})}^{2} - \frac{1}{\cos (x)} = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.9.7

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \frac{1}{\cos (x)} = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.1.1.9.8

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.2

Simplifique o lado direito.

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Etapa 80.2.1

Simplifique $(\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$ .

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Etapa 80.2.1.1

Simplifique cada termo.

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Etapa 80.2.1.1.1

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

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = ({(\frac{1}{\cos (x)})}^{2} \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.2.1.1.2

Aplique a regra do produto a $\frac{1}{\cos (x)}$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = (\frac{1^{2}}{\cos^{2} (x)} \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.2.1.1.3

Um elevado a qualquer potência é um.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = (\frac{1}{\cos^{2} (x)} \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.2.1.1.4

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

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = (\frac{1}{\cos^{2} (x)} \cdot \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.2.1.1.5

Combine.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = (\frac{1 \sin (x)}{\cos^{2} (x) \cos (x)} + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.2.1.1.6

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

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Etapa 80.2.1.1.6.1

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

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Etapa 80.2.1.1.6.1.1

Eleve $\cos (x)$ à potência de $1$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = (\frac{1 \sin (x)}{\cos^{2} (x) \cos^{1} (x)} + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.2.1.1.6.1.2

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

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = (\frac{1 \sin (x)}{{\cos (x)}^{2 + 1}} + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.2.1.1.6.2

Some $2$ e $1$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = (\frac{1 \sin (x)}{\cos^{3} (x)} + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.2.1.1.7

Multiplique $\sin (x)$ por $1$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = (\frac{\sin (x)}{\cos^{3} (x)} + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Etapa 80.2.1.2

Expanda $(\frac{\sin (x)}{\cos^{3} (x)} + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$ multiplicando cada termo na primeira expressão por cada um dos termos na segunda expressão.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin (x)}{\cos^{3} (x)} \sin (x) + \frac{\sin (x)}{\cos^{3} (x)} \cdot - 1 + \frac{\sin (x)}{\cos^{3} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Etapa 80.2.1.3

Simplifique os termos.

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Etapa 80.2.1.3.1

Simplifique cada termo.

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Etapa 80.2.1.3.1.1

Multiplique $\frac{\sin (x)}{\cos^{3} (x)} \sin (x)$ .

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Etapa 80.2.1.3.1.1.1

Combine $\frac{\sin (x)}{\cos^{3} (x)}$ e $\sin (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin (x) \sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{3} (x)} \cdot - 1 + \frac{\sin (x)}{\cos^{3} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Etapa 80.2.1.3.1.1.2

Eleve $\sin (x)$ à potência de $1$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{1} (x) \sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{3} (x)} \cdot - 1 + \frac{\sin (x)}{\cos^{3} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Etapa 80.2.1.3.1.1.3

Eleve $\sin (x)$ à potência de $1$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{1} (x) \sin^{1} (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{3} (x)} \cdot - 1 + \frac{\sin (x)}{\cos^{3} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Etapa 80.2.1.3.1.1.4

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

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{{\sin (x)}^{1 + 1}}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{3} (x)} \cdot - 1 + \frac{\sin (x)}{\cos^{3} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Etapa 80.2.1.3.1.1.5

Some $1$ e $1$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{3} (x)} \cdot - 1 + \frac{\sin (x)}{\cos^{3} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Etapa 80.2.1.3.1.2

Combine $\frac{\sin (x)}{\cos^{3} (x)}$ e $- 1$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} + \frac{\sin (x) \cdot - 1}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{3} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Etapa 80.2.1.3.1.3

Mova $- 1$ para a esquerda de $\sin (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} + \frac{- 1 \cdot \sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{3} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Etapa 80.2.1.3.1.4

Mova o número negativo para a frente da fração.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} - \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{3} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Etapa 80.2.1.3.1.5

Cancele o fator comum de $\cos (x)$ .

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Etapa 80.2.1.3.1.5.1

Fatore $\cos (x)$ de $\cos^{3} (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} - \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos (x) \cos^{2} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Etapa 80.2.1.3.1.5.2

Cancele o fator comum.

Etapa 80.2.1.3.1.5.3

Reescreva a expressão.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} - \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Etapa 80.2.1.3.1.6

Combine $\frac{1}{\cos^{3} (x)}$ e $\sin (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} - \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin (x)}{\cos^{3} (x)} + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Etapa 80.2.1.3.1.7

Combine $\frac{1}{\cos^{3} (x)}$ e $- 1$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} - \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin (x)}{\cos^{3} (x)} + \frac{- 1}{\cos^{3} (x)} + \frac{1}{\cos^{3} (x)} \cos (x)$

Etapa 80.2.1.3.1.8

Mova o número negativo para a frente da fração.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} - \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} + \frac{1}{\cos^{3} (x)} \cos (x)$

Etapa 80.2.1.3.1.9

Cancele o fator comum de $\cos (x)$ .

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Etapa 80.2.1.3.1.9.1

Fatore $\cos (x)$ de $\cos^{3} (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} - \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} + \frac{1}{\cos (x) \cos^{2} (x)} \cos (x)$

Etapa 80.2.1.3.1.9.2

Cancele o fator comum.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} - \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} + \frac{1}{\cos (x) \cos^{2} (x)} \cos (x)$

Etapa 80.2.1.3.1.9.3

Reescreva a expressão.

Etapa 80.2.1.3.2

Simplifique os termos.

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Etapa 80.2.1.3.2.1

Combine os termos opostos em $\frac{\sin^{2} (x)}{\cos^{3} (x)} - \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} + \frac{1}{\cos^{2} (x)}$ .

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Etapa 80.2.1.3.2.1.1

Some $- \frac{\sin (x)}{\cos^{3} (x)}$ e $\frac{\sin (x)}{\cos^{3} (x)}$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + 0 - \frac{1}{\cos^{3} (x)} + \frac{1}{\cos^{2} (x)}$

Etapa 80.2.1.3.2.1.2

Some $\frac{\sin^{2} (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)}$ e $0$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} - \frac{1}{\cos^{3} (x)} + \frac{1}{\cos^{2} (x)}$

Etapa 80.2.1.3.2.2

Combine os numeradores em relação ao denominador comum.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x) - 1}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Etapa 80.2.1.3.2.3

Reordene $\sin^{2} (x)$ e $- 1$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{- 1 + \sin^{2} (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Etapa 80.2.1.3.2.4

Reescreva $- 1$ como $- 1 (1)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{- 1 (1) + \sin^{2} (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Etapa 80.2.1.3.2.5

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

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{- 1 (1) - 1 (- \sin^{2} (x))}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Etapa 80.2.1.3.2.6

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

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{- 1 (1 - \sin^{2} (x))}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Etapa 80.2.1.3.2.7

Reescreva $- 1 (1 - \sin^{2} (x))$ como $- (1 - \sin^{2} (x))$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{- (1 - \sin^{2} (x))}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Etapa 80.2.1.4

Aplique a identidade trigonométrica fundamental.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{- \cos^{2} (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Etapa 80.2.1.5

Simplifique os termos.

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Etapa 80.2.1.5.1

Simplifique cada termo.

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Etapa 80.2.1.5.1.1

Cancele o fator comum de $\cos^{2} (x)$ e $\cos^{3} (x)$ .

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Etapa 80.2.1.5.1.1.1

Fatore $\cos^{2} (x)$ de $- \cos^{2} (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\cos^{2} (x) \cdot - 1}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Etapa 80.2.1.5.1.1.2

Cancele os fatores comuns.

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Etapa 80.2.1.5.1.1.2.1

Fatore $\cos^{2} (x)$ de $\cos^{3} (x)$ .

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

Etapa 80.2.1.5.1.1.2.2

Cancele o fator comum.

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

Etapa 80.2.1.5.1.1.2.3

Reescreva a expressão.

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

Etapa 80.2.1.5.1.2

Mova o número negativo para a frente da fração.

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

Etapa 80.2.1.5.2

Simplifique cada termo.

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Etapa 80.2.1.5.2.1

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

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

Etapa 80.2.1.5.2.2

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

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

Etapa 80.2.1.5.2.3

Separe as frações.

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

Etapa 80.2.1.5.2.4

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = - \sec (x) + \frac{1}{\cos (x)} \tan (x) + \frac{1}{\cos^{2} (x)}$

Etapa 80.2.1.5.2.5

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = - \sec (x) + \sec (x) \tan (x) + \frac{1}{\cos^{2} (x)}$

Etapa 80.2.1.5.2.6

Reescreva $1$ como $1^{2}$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = - \sec (x) + \sec (x) \tan (x) + \frac{1^{2}}{\cos^{2} (x)}$

Etapa 80.2.1.5.2.7

Reescreva $\frac{1^{2}}{\cos^{2} (x)}$ como ${(\frac{1}{\cos (x)})}^{2}$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = - \sec (x) + \sec (x) \tan (x) + {(\frac{1}{\cos (x)})}^{2}$

Etapa 80.2.1.5.2.8

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = - \sec (x) + \sec (x) \tan (x) + \sec^{2} (x)$

Etapa 81

Resolva $x$ .

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Etapa 81.1

Simplifique o lado esquerdo.

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Etapa 81.1.1

Simplifique cada termo.

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Etapa 81.1.1.1

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

$\frac{1}{\cos (x)} \tan (x) + \sec^{2} (x) - \sec (x) = - \sec (x) + \sec (x) \tan (x) + \sec^{2} (x)$

Etapa 81.1.1.2

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

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

Etapa 81.1.1.3

Multiplique $\frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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Etapa 81.1.1.3.1

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

$\frac{\sin (x)}{\cos (x) \cos (x)} + \sec^{2} (x) - \sec (x) = - \sec (x) + \sec (x) \tan (x) + \sec^{2} (x)$

Etapa 81.1.1.3.2

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 81.1.1.3.3

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 81.1.1.3.4

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

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

Etapa 81.1.1.3.5

Some $1$ e $1$ .

$\frac{\sin (x)}{\cos^{2} (x)} + \sec^{2} (x) - \sec (x) = - \sec (x) + \sec (x) \tan (x) + \sec^{2} (x)$

Etapa 81.1.1.4

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

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

Etapa 81.1.1.5

Aplique a regra do produto a $\frac{1}{\cos (x)}$ .

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

Etapa 81.1.1.6

Um elevado a qualquer potência é um.

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

Etapa 81.1.1.7

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

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

Etapa 81.2

Simplifique o lado direito.

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Etapa 81.2.1

Simplifique cada termo.

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Etapa 81.2.1.1

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

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

Etapa 81.2.1.2

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

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

Etapa 81.2.1.3

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

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

Etapa 81.2.1.4

Multiplique $\frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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Etapa 81.2.1.4.1

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

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

Etapa 81.2.1.4.2

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 81.2.1.4.3

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 81.2.1.4.4

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

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

Etapa 81.2.1.4.5

Some $1$ e $1$ .

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

Etapa 81.2.1.5

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

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

Etapa 81.2.1.6

Aplique a regra do produto a $\frac{1}{\cos (x)}$ .

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

Etapa 81.2.1.7

Um elevado a qualquer potência é um.

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

Etapa 81.3

Multiplique os dois lados da equação por $\cos (x)$ .

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

Etapa 81.4

Aplique a propriedade distributiva.

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

Etapa 81.5

Simplifique.

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Etapa 81.5.1

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.5.1.1

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

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

Etapa 81.5.1.2

Cancele o fator comum.

Etapa 81.5.1.3

Reescreva a expressão.

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

Etapa 81.5.2

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.5.2.1

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

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

Etapa 81.5.2.2

Cancele o fator comum.

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

Etapa 81.5.2.3

Reescreva a expressão.

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

Etapa 81.5.3

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 81.6

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.6.1

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

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

Etapa 81.6.2

Cancele o fator comum.

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

Etapa 81.6.3

Reescreva a expressão.

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

Etapa 81.7

Aplique a propriedade distributiva.

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

Etapa 81.8

Simplifique.

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Etapa 81.8.1

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 81.8.2

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.8.2.1

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

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

Etapa 81.8.2.2

Cancele o fator comum.

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

Etapa 81.8.2.3

Reescreva a expressão.

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

Etapa 81.8.3

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.8.3.1

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

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

Etapa 81.8.3.2

Cancele o fator comum.

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

Etapa 81.8.3.3

Reescreva a expressão.

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

Etapa 81.9

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.9.1

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

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

Etapa 81.9.2

Cancele o fator comum.

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

Etapa 81.9.3

Reescreva a expressão.

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

Etapa 81.10

Multiplique os dois lados da equação por $\cos (x)$ .

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

Etapa 81.11

Aplique a propriedade distributiva.

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

Etapa 81.12

Simplifique.

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Etapa 81.12.1

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.12.1.1

Cancele o fator comum.

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

Etapa 81.12.1.2

Reescreva a expressão.

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

Etapa 81.12.2

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.12.2.1

Cancele o fator comum.

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

Etapa 81.12.2.2

Reescreva a expressão.

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

Etapa 81.12.3

Mova $- 1$ para a esquerda de $\cos (x)$ .

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

Etapa 81.13

Reescreva $- 1 \cos (x)$ como $- \cos (x)$ .

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

Etapa 81.14

Aplique a propriedade distributiva.

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

Etapa 81.15

Simplifique.

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Etapa 81.15.1

Mova $- 1$ para a esquerda de $\cos (x)$ .

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

Etapa 81.15.2

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.15.2.1

Cancele o fator comum.

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

Etapa 81.15.2.2

Reescreva a expressão.

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

Etapa 81.15.3

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.15.3.1

Cancele o fator comum.

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

Etapa 81.15.3.2

Reescreva a expressão.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \sin (x) + 1$

Etapa 81.16

Reescreva $- 1 \cos (x)$ como $- \cos (x)$ .

$\sin (x) + 1 - \cos (x) = - \cos (x) + \sin (x) + 1$

Etapa 81.17

Divida cada termo na equação por $\cos (x)$ .

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

Etapa 81.18

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

$\tan (x) + \frac{1}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.19

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\tan (x) + \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.20

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.20.1

Cancele o fator comum.

$\tan (x) + \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.20.2

Divida $- 1$ por $1$ .

$\tan (x) + \sec (x) - 1 = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.21

Simplifique cada termo.

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Etapa 81.21.1

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.21.1.1

Cancele o fator comum.

$\tan (x) + \sec (x) - 1 = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.21.1.2

Divida $- 1$ por $1$ .

$\tan (x) + \sec (x) - 1 = - 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.21.2

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

$\tan (x) + \sec (x) - 1 = - 1 + \tan (x) + \frac{1}{\cos (x)}$

Etapa 81.21.3

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\tan (x) + \sec (x) - 1 = - 1 + \tan (x) + \sec (x)$

Etapa 81.22

Simplifique o lado esquerdo.

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Etapa 81.22.1

Simplifique cada termo.

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Etapa 81.22.1.1

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

$\frac{\sin (x)}{\cos (x)} + \sec (x) - 1 = - 1 + \tan (x) + \sec (x)$

Etapa 81.22.1.2

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

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - 1 + \tan (x) + \sec (x)$

Etapa 81.23

Simplifique o lado direito.

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Etapa 81.23.1

Simplifique cada termo.

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Etapa 81.23.1.1

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

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - 1 + \frac{\sin (x)}{\cos (x)} + \sec (x)$

Etapa 81.23.1.2

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

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

Etapa 81.24

Multiplique os dois lados da equação por $\cos (x)$ .

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

Etapa 81.25

Aplique a propriedade distributiva.

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

Etapa 81.26

Simplifique.

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Etapa 81.26.1

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.26.1.1

Cancele o fator comum.

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

Etapa 81.26.1.2

Reescreva a expressão.

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

Etapa 81.26.2

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.26.2.1

Cancele o fator comum.

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

Etapa 81.26.2.2

Reescreva a expressão.

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

Etapa 81.26.3

Mova $- 1$ para a esquerda de $\cos (x)$ .

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

Etapa 81.27

Reescreva $- 1 \cos (x)$ como $- \cos (x)$ .

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

Etapa 81.28

Aplique a propriedade distributiva.

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

Etapa 81.29

Simplifique.

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Etapa 81.29.1

Mova $- 1$ para a esquerda de $\cos (x)$ .

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

Etapa 81.29.2

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.29.2.1

Cancele o fator comum.

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

Etapa 81.29.2.2

Reescreva a expressão.

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

Etapa 81.29.3

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.29.3.1

Cancele o fator comum.

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

Etapa 81.29.3.2

Reescreva a expressão.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \sin (x) + 1$

Etapa 81.30

Reescreva $- 1 \cos (x)$ como $- \cos (x)$ .

$\sin (x) + 1 - \cos (x) = - \cos (x) + \sin (x) + 1$

Etapa 81.31

Divida cada termo na equação por $\cos (x)$ .

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

Etapa 81.32

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

$\tan (x) + \frac{1}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.33

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\tan (x) + \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.34

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.34.1

Cancele o fator comum.

$\tan (x) + \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.34.2

Divida $- 1$ por $1$ .

$\tan (x) + \sec (x) - 1 = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.35

Simplifique cada termo.

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Etapa 81.35.1

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.35.1.1

Cancele o fator comum.

$\tan (x) + \sec (x) - 1 = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.35.1.2

Divida $- 1$ por $1$ .

$\tan (x) + \sec (x) - 1 = - 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.35.2

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

$\tan (x) + \sec (x) - 1 = - 1 + \tan (x) + \frac{1}{\cos (x)}$

Etapa 81.35.3

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\tan (x) + \sec (x) - 1 = - 1 + \tan (x) + \sec (x)$

Etapa 81.36

Simplifique o lado esquerdo.

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Etapa 81.36.1

Simplifique cada termo.

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Etapa 81.36.1.1

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

$\frac{\sin (x)}{\cos (x)} + \sec (x) - 1 = - 1 + \tan (x) + \sec (x)$

Etapa 81.36.1.2

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

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - 1 + \tan (x) + \sec (x)$

Etapa 81.37

Simplifique o lado direito.

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Etapa 81.37.1

Simplifique cada termo.

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Etapa 81.37.1.1

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

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - 1 + \frac{\sin (x)}{\cos (x)} + \sec (x)$

Etapa 81.37.1.2

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

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

Etapa 81.38

Multiplique os dois lados da equação por $\cos (x)$ .

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

Etapa 81.39

Aplique a propriedade distributiva.

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

Etapa 81.40

Simplifique.

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Etapa 81.40.1

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.40.1.1

Cancele o fator comum.

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

Etapa 81.40.1.2

Reescreva a expressão.

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

Etapa 81.40.2

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.40.2.1

Cancele o fator comum.

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

Etapa 81.40.2.2

Reescreva a expressão.

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

Etapa 81.40.3

Mova $- 1$ para a esquerda de $\cos (x)$ .

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

Etapa 81.41

Reescreva $- 1 \cos (x)$ como $- \cos (x)$ .

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

Etapa 81.42

Aplique a propriedade distributiva.

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

Etapa 81.43

Simplifique.

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Etapa 81.43.1

Mova $- 1$ para a esquerda de $\cos (x)$ .

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

Etapa 81.43.2

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.43.2.1

Cancele o fator comum.

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

Etapa 81.43.2.2

Reescreva a expressão.

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

Etapa 81.43.3

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.43.3.1

Cancele o fator comum.

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

Etapa 81.43.3.2

Reescreva a expressão.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \sin (x) + 1$

Etapa 81.44

Reescreva $- 1 \cos (x)$ como $- \cos (x)$ .

$\sin (x) + 1 - \cos (x) = - \cos (x) + \sin (x) + 1$

Etapa 81.45

Divida cada termo na equação por $\cos (x)$ .

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

Etapa 81.46

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

$\tan (x) + \frac{1}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.47

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\tan (x) + \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.48

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.48.1

Cancele o fator comum.

$\tan (x) + \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.48.2

Divida $- 1$ por $1$ .

$\tan (x) + \sec (x) - 1 = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.49

Simplifique cada termo.

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Etapa 81.49.1

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.49.1.1

Cancele o fator comum.

$\tan (x) + \sec (x) - 1 = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.49.1.2

Divida $- 1$ por $1$ .

$\tan (x) + \sec (x) - 1 = - 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.49.2

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

$\tan (x) + \sec (x) - 1 = - 1 + \tan (x) + \frac{1}{\cos (x)}$

Etapa 81.49.3

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\tan (x) + \sec (x) - 1 = - 1 + \tan (x) + \sec (x)$

Etapa 81.50

Simplifique o lado esquerdo.

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Etapa 81.50.1

Simplifique cada termo.

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Etapa 81.50.1.1

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

$\frac{\sin (x)}{\cos (x)} + \sec (x) - 1 = - 1 + \tan (x) + \sec (x)$

Etapa 81.50.1.2

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

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - 1 + \tan (x) + \sec (x)$

Etapa 81.51

Simplifique o lado direito.

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Etapa 81.51.1

Simplifique cada termo.

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Etapa 81.51.1.1

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

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - 1 + \frac{\sin (x)}{\cos (x)} + \sec (x)$

Etapa 81.51.1.2

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

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

Etapa 81.52

Multiplique os dois lados da equação por $\cos (x)$ .

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

Etapa 81.53

Aplique a propriedade distributiva.

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

Etapa 81.54

Simplifique.

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Etapa 81.54.1

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.54.1.1

Cancele o fator comum.

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

Etapa 81.54.1.2

Reescreva a expressão.

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

Etapa 81.54.2

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.54.2.1

Cancele o fator comum.

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

Etapa 81.54.2.2

Reescreva a expressão.

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

Etapa 81.54.3

Mova $- 1$ para a esquerda de $\cos (x)$ .

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

Etapa 81.55

Reescreva $- 1 \cos (x)$ como $- \cos (x)$ .

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

Etapa 81.56

Aplique a propriedade distributiva.

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

Etapa 81.57

Simplifique.

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Etapa 81.57.1

Mova $- 1$ para a esquerda de $\cos (x)$ .

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

Etapa 81.57.2

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.57.2.1

Cancele o fator comum.

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

Etapa 81.57.2.2

Reescreva a expressão.

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

Etapa 81.57.3

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.57.3.1

Cancele o fator comum.

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

Etapa 81.57.3.2

Reescreva a expressão.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \sin (x) + 1$

Etapa 81.58

Reescreva $- 1 \cos (x)$ como $- \cos (x)$ .

$\sin (x) + 1 - \cos (x) = - \cos (x) + \sin (x) + 1$

Etapa 81.59

Divida cada termo na equação por $\cos (x)$ .

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

Etapa 81.60

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

$\tan (x) + \frac{1}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.61

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\tan (x) + \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.62

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.62.1

Cancele o fator comum.

$\tan (x) + \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.62.2

Divida $- 1$ por $1$ .

$\tan (x) + \sec (x) - 1 = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.63

Simplifique cada termo.

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Etapa 81.63.1

Cancele o fator comum de $\cos (x)$ .

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Etapa 81.63.1.1

Cancele o fator comum.

$\tan (x) + \sec (x) - 1 = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.63.1.2

Divida $- 1$ por $1$ .

$\tan (x) + \sec (x) - 1 = - 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Etapa 81.63.2

Converta de $\frac{\sin (x)}{\cos (x)}$ em $\tan (x)$ .

$\tan (x) + \sec (x) - 1 = - 1 + \tan (x) + \frac{1}{\cos (x)}$

Etapa 81.63.3

Converta de $\frac{1}{\cos (x)}$ em $\sec (x)$ .

$\tan (x) + \sec (x) - 1 = - 1 + \tan (x) + \sec (x)$

Etapa 81.64

Mova todos os termos que contêm $x$ para o lado esquerdo da equação.

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Etapa 81.64.1

Subtraia $\tan (x)$ dos dois lados da equação.

$\tan (x) + \sec (x) - 1 - \tan (x) = - 1 + \sec (x)$

Etapa 81.64.2

Subtraia $\sec (x)$ dos dois lados da equação.

$\tan (x) + \sec (x) - 1 - \tan (x) - \sec (x) = - 1$

Etapa 81.64.3

Combine os termos opostos em $\tan (x) + \sec (x) - 1 - \tan (x) - \sec (x)$ .

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Etapa 81.64.3.1

Subtraia $\tan (x)$ de $\tan (x)$ .

$0 + \sec (x) - 1 - \sec (x) = - 1$

Etapa 81.64.3.2

Some $0$ e $\sec (x)$ .

$\sec (x) - 1 - \sec (x) = - 1$

Etapa 81.64.3.3

Subtraia $\sec (x)$ de $\sec (x)$ .

$0 - 1 = - 1$

Etapa 81.64.3.4

Subtraia $1$ de $0$ .

$- 1 = - 1$

Etapa 81.65

Como $- 1 = - 1$ , a equação sempre será verdadeira para qualquer valor de $x$ .

Todos os números reais

Etapa 82

O resultado pode ser mostrado de várias formas.

Todos os números reais

Notação de intervalo:

$(- \infty, \infty)$