Pré-cálculo Exemplos

$\cot (x) \sec^{4} (x) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Etapa 1

Simplifique o lado esquerdo.

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Etapa 1.1

Simplifique $\cot (x) \sec^{4} (x)$ .

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Etapa 1.1.1

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

$\frac{\cos (x)}{\sin (x)} \sec^{4} (x) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Etapa 1.1.2

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

$\frac{\cos (x)}{\sin (x)} {(\frac{1}{\cos (x)})}^{4} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Etapa 1.1.3

Simplifique a expressão.

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Etapa 1.1.3.1

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

$\frac{\cos (x)}{\sin (x)} \cdot \frac{1^{4}}{\cos^{4} (x)} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Etapa 1.1.3.2

Um elevado a qualquer potência é um.

$\frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\cos^{4} (x)} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Etapa 1.1.4

Multiplique $\frac{\cos (x)}{\sin (x)}$ por $\frac{1}{\cos^{4} (x)}$ .

$\frac{\cos (x)}{\sin (x) \cos^{4} (x)} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Etapa 1.1.5

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

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Etapa 1.1.5.1

Multiplique por $1$ .

$\frac{\cos (x) \cdot 1}{\sin (x) \cos^{4} (x)} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Etapa 1.1.5.2

Cancele os fatores comuns.

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Etapa 1.1.5.2.1

Fatore $\cos (x)$ de $\sin (x) \cos^{4} (x)$ .

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

Etapa 1.1.5.2.2

Cancele o fator comum.

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

Etapa 1.1.5.2.3

Reescreva a expressão.

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

Etapa 1.1.6

Separe as frações.

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

Etapa 1.1.7

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

$\frac{1}{\cos^{3} (x)} \csc (x) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Etapa 1.1.8

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

$\frac{\csc (x)}{\cos^{3} (x)} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Etapa 1.1.9

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

$\frac{\csc (x)}{\cos (x) \cos^{2} (x)} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Etapa 1.1.10

Separe as frações.

$\frac{1}{\cos^{2} (x)} \cdot \frac{\csc (x)}{\cos (x)} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Etapa 1.1.11

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

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

Etapa 1.1.12

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

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

Etapa 1.1.13

Simplifique.

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Etapa 1.1.13.1

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

$\frac{1}{\cos^{2} (x)} (\csc (x) \frac{1}{\cos (x)}) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Etapa 1.1.13.2

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

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

Etapa 1.1.14

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

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

Etapa 1.1.15

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

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

Etapa 1.1.16

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

$\sec^{2} (x) (\csc (x) \sec (x)) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Etapa 1.1.17

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

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Etapa 1.1.17.1

Mova $\sec (x)$ .

$\sec (x) \sec^{2} (x) \csc (x) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Etapa 1.1.17.2

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

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Etapa 1.1.17.2.1

Eleve $\sec (x)$ à potência de $1$ .

$\sec^{1} (x) \sec^{2} (x) \csc (x) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Etapa 1.1.17.2.2

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

${\sec (x)}^{1 + 2} \csc (x) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Etapa 1.1.17.3

Some $1$ e $2$ .

$\sec^{3} (x) \csc (x) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Etapa 2

Como $x$ está do lado direito da equação, troque os lados para que ela fique do lado esquerdo da equação.

$\cot (x) + 2 \tan (x) + \tan^{3} (x) = \sec^{3} (x) \csc (x)$

Etapa 3

Simplifique o lado esquerdo.

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Etapa 3.1

Simplifique cada termo.

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Etapa 3.1.1

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

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

Etapa 3.1.2

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

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

Etapa 3.1.3

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

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

Etapa 3.1.4

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

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

Etapa 3.1.5

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

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

Etapa 4

Simplifique o lado direito.

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Etapa 4.1

Simplifique $\sec^{3} (x) \csc (x)$ .

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Etapa 4.1.1

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

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

Etapa 4.1.2

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

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

Etapa 4.1.3

Um elevado a qualquer potência é um.

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

Etapa 4.1.4

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

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

Etapa 4.1.5

Combine.

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

Etapa 4.1.6

Multiplique $1$ por $1$ .

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

Etapa 5

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

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

Etapa 6

Aplique a propriedade distributiva.

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

Etapa 7

Simplifique.

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Etapa 7.1

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

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Etapa 7.1.1

Cancele o fator comum.

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

Etapa 7.1.2

Reescreva a expressão.

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

Etapa 7.2

Multiplique $\sin (x) \frac{2 \sin (x)}{\cos (x)}$ .

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Etapa 7.2.1

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

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

Etapa 7.2.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 7.2.3

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 7.2.4

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

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

Etapa 7.2.5

Some $1$ e $1$ .

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

Etapa 7.3

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

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Etapa 7.3.1

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

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

Etapa 7.3.2

Multiplique $\sin (x)$ por $\sin^{3} (x)$ somando os expoentes.

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Etapa 7.3.2.1

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

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Etapa 7.3.2.1.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 7.3.2.1.2

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

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

Etapa 7.3.2.2

Some $1$ e $3$ .

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} = \sin (x) \frac{1}{\cos^{3} (x) \sin (x)}$

Etapa 8

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

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Etapa 8.1

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

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} = \sin (x) \frac{1}{\sin (x) \cos^{3} (x)}$

Etapa 8.2

Cancele o fator comum.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} = \sin (x) \frac{1}{\sin (x) \cos^{3} (x)}$

Etapa 8.3

Reescreva a expressão.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} = \frac{1}{\cos^{3} (x)}$

Etapa 9

Subtraia $\frac{1}{\cos^{3} (x)}$ dos dois lados da equação.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = 0$

Etapa 10

Simplifique cada termo.

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Etapa 10.1

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

$\cos (x) + \frac{2 (\sin (x) \sin (x))}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = 0$

Etapa 10.2

Separe as frações.

$\cos (x) + \frac{2 (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = 0$

Etapa 10.3

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

$\cos (x) + \frac{2 (\sin (x))}{1} \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = 0$

Etapa 10.4

Divida $2 (\sin (x))$ por $1$ .

$\cos (x) + 2 (\sin (x)) \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = 0$

Etapa 10.5

Fatore $\sin^{3} (x)$ de $\sin^{4} (x)$ .

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

Etapa 10.6

Multiplique por $1$ .

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

Etapa 10.7

Separe as frações.

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

Etapa 10.8

Converta de $\frac{\sin^{3} (x)}{\cos^{3} (x)}$ em $\tan^{3} (x)$ .

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

Etapa 10.9

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

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

Etapa 11

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

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

Etapa 12

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

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Etapa 12.1

Cancele o fator comum.

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

Etapa 12.2

Reescreva a expressão.

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

Etapa 13

Simplifique o numerador.

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Etapa 13.1

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

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

Etapa 13.2

Combine expoentes.

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Etapa 13.2.1

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

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

Etapa 13.2.2

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

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

Etapa 13.2.3

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 13.2.4

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 13.2.5

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

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

Etapa 13.2.6

Some $1$ e $1$ .

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

Etapa 14

Multiplique o numerador pelo inverso do denominador.

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

Etapa 15

Combine frações.

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Etapa 15.1

Combine.

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

Etapa 15.2

Multiplique $2$ por $1$ .

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

Etapa 16

Simplifique o denominador.

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Etapa 16.1

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

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

Etapa 16.2

Some $1$ e $1$ .

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

Etapa 17

Multiplique por $1$ .

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

Etapa 18

Multiplique por $1$ .

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

Etapa 19

Separe as frações.

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

Etapa 20

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

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

Etapa 21

Multiplique $2$ por $1$ .

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

Etapa 22

Divida $2$ por $1$ .

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

Etapa 23

Simplifique o numerador.

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Etapa 23.1

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

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

Etapa 23.2

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

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

Etapa 24

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

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

Etapa 25

Multiplique $\sin (x)$ por $\sin^{3} (x)$ somando os expoentes.

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Etapa 25.1

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

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Etapa 25.1.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 25.1.2

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

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

Etapa 25.2

Some $1$ e $3$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin^{4} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 26

Multiplique o numerador pelo inverso do denominador.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 27

Combine.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 28

Multiplique $\cos^{3} (x)$ por $\cos (x)$ somando os expoentes.

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Etapa 28.1

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

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Etapa 28.1.1

Eleve $\cos (x)$ à potência de $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 28.1.2

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

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{{\cos (x)}^{3 + 1}} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 28.2

Some $3$ e $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 29

Multiplique $\sin^{4} (x)$ por $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 30

Converta de $\frac{\sin^{4} (x)}{\cos^{4} (x)}$ em $\tan^{4} (x)$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 31

Multiplique o numerador pelo inverso do denominador.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 32

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \sec (x) = \frac{0}{\cos (x)}$

Etapa 33

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos^{3} (x)} = \frac{0}{\cos (x)}$

Etapa 34

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos (x) \cos^{2} (x)} = \frac{0}{\cos (x)}$

Etapa 35

Separe as frações.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\sec (x)}{\cos (x)}) = \frac{0}{\cos (x)}$

Etapa 36

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\frac{1}{\cos (x)}}{\cos (x)}) = \frac{0}{\cos (x)}$

Etapa 37

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)})) = \frac{0}{\cos (x)}$

Etapa 38

Simplifique.

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Etapa 38.1

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) (\frac{1}{\cos (x)}))) = \frac{0}{\cos (x)}$

Etapa 38.2

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Etapa 38.3

Eleve $\sec (x)$ à potência de $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Etapa 38.4

Eleve $\sec (x)$ à potência de $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Etapa 38.5

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot {\sec (x)}^{1 + 1}) = \frac{0}{\cos (x)}$

Etapa 38.6

Some $1$ e $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 39

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1^{2}}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 40

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - ({(\frac{1}{\cos (x)})}^{2} \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 41

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\sec^{2} (x) \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 42

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

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Etapa 42.1

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - {\sec (x)}^{2 + 2} = \frac{0}{\cos (x)}$

Etapa 42.2

Some $2$ e $2$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{\cos (x)}$

Etapa 43

Separe as frações.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Etapa 44

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \sec (x)$

Etapa 45

Divida $0$ por $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0 \sec (x)$

Etapa 46

Multiplique $0$ por $\sec (x)$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 47

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

$1 + 2 (\sec^{2} (x) - 1) + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 48

Simplifique cada termo.

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Etapa 48.1

Aplique a propriedade distributiva.

$1 + 2 \sec^{2} (x) + 2 \cdot - 1 + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 48.2

Multiplique $2$ por $- 1$ .

$1 + 2 \sec^{2} (x) - 2 + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 49

Subtraia $2$ de $1$ .

$2 \sec^{2} (x) - 1 + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 50

Reordene o polinômio.

$\tan^{4} (x) - \sec^{4} (x) + 2 \sec^{2} (x) - 1 = 0$

Etapa 51

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

$(1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52

Simplifique.

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Etapa 52.1

Simplifique o lado esquerdo.

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Etapa 52.1.1

Simplifique $(1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x)) \cos (x)$ .

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Etapa 52.1.1.1

Simplifique cada termo.

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Etapa 52.1.1.1.1

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

$(1 + 2 {(\frac{\sin (x)}{\cos (x)})}^{2} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.1.2

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

$(1 + 2 \frac{\sin^{2} (x)}{\cos^{2} (x)} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.1.3

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.1.4

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + {(\frac{\sin (x)}{\cos (x)})}^{4} - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.1.5

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.1.6

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - {(\frac{1}{\cos (x)})}^{4}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.1.7

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \frac{1^{4}}{\cos^{4} (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.1.8

Um elevado a qualquer potência é um.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \frac{1}{\cos^{4} (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.2

Aplique a propriedade distributiva.

$1 \cos (x) + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.3

Simplifique.

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Etapa 52.1.1.3.1

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

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.3.2

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

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Etapa 52.1.1.3.2.1

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

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.3.2.2

Cancele o fator comum.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.3.2.3

Reescreva a expressão.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.3.3

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

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Etapa 52.1.1.3.3.1

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

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos (x) \cos^{3} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.3.3.2

Cancele o fator comum.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos (x) \cos^{3} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.3.3.3

Reescreva a expressão.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.3.4

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

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Etapa 52.1.1.3.4.1

Mova o negativo de maior ordem em $- \frac{1}{\cos^{4} (x)}$ para o numerador.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.3.4.2

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

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos (x) \cos^{3} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.3.4.3

Cancele o fator comum.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos (x) \cos^{3} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.3.4.4

Reescreva a expressão.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.4

Mova o número negativo para a frente da fração.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.5

Simplifique cada termo.

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Etapa 52.1.1.5.1

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

$\cos (x) + \frac{2 (\sin (x) \sin (x))}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.5.2

Separe as frações.

$\cos (x) + \frac{2 (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.5.3

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

$\cos (x) + \frac{2 (\sin (x))}{1} \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.5.4

Divida $2 (\sin (x))$ por $1$ .

$\cos (x) + 2 (\sin (x)) \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 52.1.1.5.5

Fatore $\sin^{3} (x)$ de $\sin^{4} (x)$ .

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

Etapa 52.1.1.5.6

Multiplique por $1$ .

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

Etapa 52.1.1.5.7

Separe as frações.

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

Etapa 52.1.1.5.8

Converta de $\frac{\sin^{3} (x)}{\cos^{3} (x)}$ em $\tan^{3} (x)$ .

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

Etapa 52.1.1.5.9

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

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

Etapa 52.2

Simplifique o lado direito.

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Etapa 52.2.1

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

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Etapa 52.2.1.1

Cancele o fator comum.

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

Etapa 52.2.1.2

Reescreva a expressão.

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

Etapa 53

Resolva $x$ .

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Etapa 53.1

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

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

Etapa 53.2

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

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Etapa 53.2.1

Cancele o fator comum.

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

Etapa 53.2.2

Reescreva a expressão.

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

Etapa 53.3

Simplifique o numerador.

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Etapa 53.3.1

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

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

Etapa 53.3.2

Combine expoentes.

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Etapa 53.3.2.1

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

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

Etapa 53.3.2.2

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

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

Etapa 53.3.2.3

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 53.3.2.4

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 53.3.2.5

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

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

Etapa 53.3.2.6

Some $1$ e $1$ .

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

Etapa 53.4

Multiplique o numerador pelo inverso do denominador.

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

Etapa 53.5

Combine frações.

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Etapa 53.5.1

Combine.

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

Etapa 53.5.2

Multiplique $2$ por $1$ .

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

Etapa 53.6

Simplifique o denominador.

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Etapa 53.6.1

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

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

Etapa 53.6.2

Some $1$ e $1$ .

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

Etapa 53.7

Multiplique por $1$ .

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

Etapa 53.8

Multiplique por $1$ .

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

Etapa 53.9

Separe as frações.

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

Etapa 53.10

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

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

Etapa 53.11

Multiplique $2$ por $1$ .

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

Etapa 53.12

Divida $2$ por $1$ .

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

Etapa 53.13

Simplifique o numerador.

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Etapa 53.13.1

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

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

Etapa 53.13.2

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

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

Etapa 53.14

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

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

Etapa 53.15

Multiplique $\sin (x)$ por $\sin^{3} (x)$ somando os expoentes.

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Etapa 53.15.1

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

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Etapa 53.15.1.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 53.15.1.2

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

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

Etapa 53.15.2

Some $1$ e $3$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin^{4} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.16

Multiplique o numerador pelo inverso do denominador.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.17

Combine.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.18

Multiplique $\cos^{3} (x)$ por $\cos (x)$ somando os expoentes.

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Etapa 53.18.1

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

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Etapa 53.18.1.1

Eleve $\cos (x)$ à potência de $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.18.1.2

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

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{{\cos (x)}^{3 + 1}} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.18.2

Some $3$ e $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.19

Multiplique $\sin^{4} (x)$ por $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.20

Converta de $\frac{\sin^{4} (x)}{\cos^{4} (x)}$ em $\tan^{4} (x)$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.21

Multiplique o numerador pelo inverso do denominador.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.22

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \sec (x) = \frac{0}{\cos (x)}$

Etapa 53.23

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos^{3} (x)} = \frac{0}{\cos (x)}$

Etapa 53.24

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos (x) \cos^{2} (x)} = \frac{0}{\cos (x)}$

Etapa 53.25

Separe as frações.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\sec (x)}{\cos (x)}) = \frac{0}{\cos (x)}$

Etapa 53.26

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\frac{1}{\cos (x)}}{\cos (x)}) = \frac{0}{\cos (x)}$

Etapa 53.27

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)})) = \frac{0}{\cos (x)}$

Etapa 53.28

Simplifique.

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Etapa 53.28.1

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) (\frac{1}{\cos (x)}))) = \frac{0}{\cos (x)}$

Etapa 53.28.2

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Etapa 53.28.3

Eleve $\sec (x)$ à potência de $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Etapa 53.28.4

Eleve $\sec (x)$ à potência de $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Etapa 53.28.5

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot {\sec (x)}^{1 + 1}) = \frac{0}{\cos (x)}$

Etapa 53.28.6

Some $1$ e $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 53.29

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1^{2}}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 53.30

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - ({(\frac{1}{\cos (x)})}^{2} \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 53.31

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\sec^{2} (x) \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 53.32

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

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Etapa 53.32.1

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - {\sec (x)}^{2 + 2} = \frac{0}{\cos (x)}$

Etapa 53.32.2

Some $2$ e $2$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{\cos (x)}$

Etapa 53.33

Separe as frações.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Etapa 53.34

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \sec (x)$

Etapa 53.35

Divida $0$ por $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0 \sec (x)$

Etapa 53.36

Multiplique $0$ por $\sec (x)$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 53.37

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

$1 + 2 (\sec^{2} (x) - 1) + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 53.38

Simplifique cada termo.

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Etapa 53.38.1

Aplique a propriedade distributiva.

$1 + 2 \sec^{2} (x) + 2 \cdot - 1 + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 53.38.2

Multiplique $2$ por $- 1$ .

$1 + 2 \sec^{2} (x) - 2 + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 53.39

Subtraia $2$ de $1$ .

$2 \sec^{2} (x) - 1 + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 53.40

Reordene o polinômio.

$\tan^{4} (x) - \sec^{4} (x) + 2 \sec^{2} (x) - 1 = 0$

Etapa 53.41

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

$(1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42

Simplifique.

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Etapa 53.42.1

Simplifique o lado esquerdo.

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Etapa 53.42.1.1

Simplifique $(1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x)) \cos (x)$ .

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Etapa 53.42.1.1.1

Simplifique cada termo.

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Etapa 53.42.1.1.1.1

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

$(1 + 2 {(\frac{\sin (x)}{\cos (x)})}^{2} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.1.2

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

$(1 + 2 \frac{\sin^{2} (x)}{\cos^{2} (x)} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.1.3

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.1.4

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + {(\frac{\sin (x)}{\cos (x)})}^{4} - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.1.5

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.1.6

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - {(\frac{1}{\cos (x)})}^{4}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.1.7

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \frac{1^{4}}{\cos^{4} (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.1.8

Um elevado a qualquer potência é um.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \frac{1}{\cos^{4} (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.2

Aplique a propriedade distributiva.

$1 \cos (x) + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.3

Simplifique.

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Etapa 53.42.1.1.3.1

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

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.3.2

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

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Etapa 53.42.1.1.3.2.1

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

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.3.2.2

Cancele o fator comum.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.3.2.3

Reescreva a expressão.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.3.3

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

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Etapa 53.42.1.1.3.3.1

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

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos (x) \cos^{3} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.3.3.2

Cancele o fator comum.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos (x) \cos^{3} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.3.3.3

Reescreva a expressão.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.3.4

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

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Etapa 53.42.1.1.3.4.1

Mova o negativo de maior ordem em $- \frac{1}{\cos^{4} (x)}$ para o numerador.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.3.4.2

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

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos (x) \cos^{3} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.3.4.3

Cancele o fator comum.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos (x) \cos^{3} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.3.4.4

Reescreva a expressão.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.4

Mova o número negativo para a frente da fração.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.5

Simplifique cada termo.

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Etapa 53.42.1.1.5.1

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

$\cos (x) + \frac{2 (\sin (x) \sin (x))}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.5.2

Separe as frações.

$\cos (x) + \frac{2 (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.5.3

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

$\cos (x) + \frac{2 (\sin (x))}{1} \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.5.4

Divida $2 (\sin (x))$ por $1$ .

$\cos (x) + 2 (\sin (x)) \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.42.1.1.5.5

Fatore $\sin^{3} (x)$ de $\sin^{4} (x)$ .

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

Etapa 53.42.1.1.5.6

Multiplique por $1$ .

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

Etapa 53.42.1.1.5.7

Separe as frações.

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

Etapa 53.42.1.1.5.8

Converta de $\frac{\sin^{3} (x)}{\cos^{3} (x)}$ em $\tan^{3} (x)$ .

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

Etapa 53.42.1.1.5.9

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

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

Etapa 53.42.2

Simplifique o lado direito.

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Etapa 53.42.2.1

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

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Etapa 53.42.2.1.1

Cancele o fator comum.

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

Etapa 53.42.2.1.2

Reescreva a expressão.

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

Etapa 53.43

Resolva $x$ .

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Etapa 53.43.1

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

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

Etapa 53.43.2

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

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Etapa 53.43.2.1

Cancele o fator comum.

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

Etapa 53.43.2.2

Reescreva a expressão.

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

Etapa 53.43.3

Simplifique o numerador.

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Etapa 53.43.3.1

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

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

Etapa 53.43.3.2

Combine expoentes.

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Etapa 53.43.3.2.1

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

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

Etapa 53.43.3.2.2

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

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

Etapa 53.43.3.2.3

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 53.43.3.2.4

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 53.43.3.2.5

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

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

Etapa 53.43.3.2.6

Some $1$ e $1$ .

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

Etapa 53.43.4

Multiplique o numerador pelo inverso do denominador.

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

Etapa 53.43.5

Combine frações.

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Etapa 53.43.5.1

Combine.

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

Etapa 53.43.5.2

Multiplique $2$ por $1$ .

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

Etapa 53.43.6

Simplifique o denominador.

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Etapa 53.43.6.1

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

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

Etapa 53.43.6.2

Some $1$ e $1$ .

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

Etapa 53.43.7

Multiplique por $1$ .

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

Etapa 53.43.8

Multiplique por $1$ .

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

Etapa 53.43.9

Separe as frações.

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

Etapa 53.43.10

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

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

Etapa 53.43.11

Multiplique $2$ por $1$ .

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

Etapa 53.43.12

Divida $2$ por $1$ .

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

Etapa 53.43.13

Simplifique o numerador.

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Etapa 53.43.13.1

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

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

Etapa 53.43.13.2

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

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

Etapa 53.43.14

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

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

Etapa 53.43.15

Multiplique $\sin (x)$ por $\sin^{3} (x)$ somando os expoentes.

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Etapa 53.43.15.1

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

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Etapa 53.43.15.1.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 53.43.15.1.2

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

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

Etapa 53.43.15.2

Some $1$ e $3$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin^{4} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.16

Multiplique o numerador pelo inverso do denominador.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.17

Combine.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.18

Multiplique $\cos^{3} (x)$ por $\cos (x)$ somando os expoentes.

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Etapa 53.43.18.1

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

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Etapa 53.43.18.1.1

Eleve $\cos (x)$ à potência de $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.18.1.2

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

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{{\cos (x)}^{3 + 1}} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.18.2

Some $3$ e $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.19

Multiplique $\sin^{4} (x)$ por $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.20

Converta de $\frac{\sin^{4} (x)}{\cos^{4} (x)}$ em $\tan^{4} (x)$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.21

Multiplique o numerador pelo inverso do denominador.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.22

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \sec (x) = \frac{0}{\cos (x)}$

Etapa 53.43.23

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos^{3} (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.24

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos (x) \cos^{2} (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.25

Separe as frações.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\sec (x)}{\cos (x)}) = \frac{0}{\cos (x)}$

Etapa 53.43.26

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\frac{1}{\cos (x)}}{\cos (x)}) = \frac{0}{\cos (x)}$

Etapa 53.43.27

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)})) = \frac{0}{\cos (x)}$

Etapa 53.43.28

Simplifique.

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Etapa 53.43.28.1

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) (\frac{1}{\cos (x)}))) = \frac{0}{\cos (x)}$

Etapa 53.43.28.2

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Etapa 53.43.28.3

Eleve $\sec (x)$ à potência de $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Etapa 53.43.28.4

Eleve $\sec (x)$ à potência de $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Etapa 53.43.28.5

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot {\sec (x)}^{1 + 1}) = \frac{0}{\cos (x)}$

Etapa 53.43.28.6

Some $1$ e $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 53.43.29

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1^{2}}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 53.43.30

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - ({(\frac{1}{\cos (x)})}^{2} \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 53.43.31

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\sec^{2} (x) \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 53.43.32

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

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Etapa 53.43.32.1

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - {\sec (x)}^{2 + 2} = \frac{0}{\cos (x)}$

Etapa 53.43.32.2

Some $2$ e $2$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{\cos (x)}$

Etapa 53.43.33

Separe as frações.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Etapa 53.43.34

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \sec (x)$

Etapa 53.43.35

Divida $0$ por $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0 \sec (x)$

Etapa 53.43.36

Multiplique $0$ por $\sec (x)$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 53.43.37

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

$1 + 2 (\sec^{2} (x) - 1) + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 53.43.38

Simplifique cada termo.

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Etapa 53.43.38.1

Aplique a propriedade distributiva.

$1 + 2 \sec^{2} (x) + 2 \cdot - 1 + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 53.43.38.2

Multiplique $2$ por $- 1$ .

$1 + 2 \sec^{2} (x) - 2 + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 53.43.39

Subtraia $2$ de $1$ .

$2 \sec^{2} (x) - 1 + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 53.43.40

Reordene o polinômio.

$\tan^{4} (x) - \sec^{4} (x) + 2 \sec^{2} (x) - 1 = 0$

Etapa 53.43.41

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

$(1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42

Simplifique.

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Etapa 53.43.42.1

Simplifique o lado esquerdo.

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Etapa 53.43.42.1.1

Simplifique $(1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x)) \cos (x)$ .

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Etapa 53.43.42.1.1.1

Simplifique cada termo.

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Etapa 53.43.42.1.1.1.1

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

$(1 + 2 {(\frac{\sin (x)}{\cos (x)})}^{2} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.1.2

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

$(1 + 2 \frac{\sin^{2} (x)}{\cos^{2} (x)} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.1.3

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.1.4

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + {(\frac{\sin (x)}{\cos (x)})}^{4} - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.1.5

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.1.6

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - {(\frac{1}{\cos (x)})}^{4}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.1.7

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \frac{1^{4}}{\cos^{4} (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.1.8

Um elevado a qualquer potência é um.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \frac{1}{\cos^{4} (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.2

Aplique a propriedade distributiva.

$1 \cos (x) + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.3

Simplifique.

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Etapa 53.43.42.1.1.3.1

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

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.3.2

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

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Etapa 53.43.42.1.1.3.2.1

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

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.3.2.2

Cancele o fator comum.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.3.2.3

Reescreva a expressão.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.3.3

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

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Etapa 53.43.42.1.1.3.3.1

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

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos (x) \cos^{3} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.3.3.2

Cancele o fator comum.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos (x) \cos^{3} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.3.3.3

Reescreva a expressão.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.3.4

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

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Etapa 53.43.42.1.1.3.4.1

Mova o negativo de maior ordem em $- \frac{1}{\cos^{4} (x)}$ para o numerador.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.3.4.2

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

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos (x) \cos^{3} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.3.4.3

Cancele o fator comum.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos (x) \cos^{3} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.3.4.4

Reescreva a expressão.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.4

Mova o número negativo para a frente da fração.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.5

Simplifique cada termo.

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Etapa 53.43.42.1.1.5.1

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

$\cos (x) + \frac{2 (\sin (x) \sin (x))}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.5.2

Separe as frações.

$\cos (x) + \frac{2 (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.5.3

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

$\cos (x) + \frac{2 (\sin (x))}{1} \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.5.4

Divida $2 (\sin (x))$ por $1$ .

$\cos (x) + 2 (\sin (x)) \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.42.1.1.5.5

Fatore $\sin^{3} (x)$ de $\sin^{4} (x)$ .

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

Etapa 53.43.42.1.1.5.6

Multiplique por $1$ .

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

Etapa 53.43.42.1.1.5.7

Separe as frações.

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

Etapa 53.43.42.1.1.5.8

Converta de $\frac{\sin^{3} (x)}{\cos^{3} (x)}$ em $\tan^{3} (x)$ .

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

Etapa 53.43.42.1.1.5.9

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

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

Etapa 53.43.42.2

Simplifique o lado direito.

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Etapa 53.43.42.2.1

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

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Etapa 53.43.42.2.1.1

Cancele o fator comum.

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

Etapa 53.43.42.2.1.2

Reescreva a expressão.

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

Etapa 53.43.43

Resolva $x$ .

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Etapa 53.43.43.1

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

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

Etapa 53.43.43.2

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

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Etapa 53.43.43.2.1

Cancele o fator comum.

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

Etapa 53.43.43.2.2

Reescreva a expressão.

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

Etapa 53.43.43.3

Simplifique o numerador.

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Etapa 53.43.43.3.1

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

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

Etapa 53.43.43.3.2

Combine expoentes.

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Etapa 53.43.43.3.2.1

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

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

Etapa 53.43.43.3.2.2

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

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

Etapa 53.43.43.3.2.3

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 53.43.43.3.2.4

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 53.43.43.3.2.5

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

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

Etapa 53.43.43.3.2.6

Some $1$ e $1$ .

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

Etapa 53.43.43.4

Multiplique o numerador pelo inverso do denominador.

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

Etapa 53.43.43.5

Combine frações.

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Etapa 53.43.43.5.1

Combine.

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

Etapa 53.43.43.5.2

Multiplique $2$ por $1$ .

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

Etapa 53.43.43.6

Simplifique o denominador.

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Etapa 53.43.43.6.1

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

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

Etapa 53.43.43.6.2

Some $1$ e $1$ .

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

Etapa 53.43.43.7

Multiplique por $1$ .

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

Etapa 53.43.43.8

Multiplique por $1$ .

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

Etapa 53.43.43.9

Separe as frações.

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

Etapa 53.43.43.10

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

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

Etapa 53.43.43.11

Multiplique $2$ por $1$ .

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

Etapa 53.43.43.12

Divida $2$ por $1$ .

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

Etapa 53.43.43.13

Simplifique o numerador.

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Etapa 53.43.43.13.1

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

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

Etapa 53.43.43.13.2

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

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

Etapa 53.43.43.14

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

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

Etapa 53.43.43.15

Multiplique $\sin (x)$ por $\sin^{3} (x)$ somando os expoentes.

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Etapa 53.43.43.15.1

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

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Etapa 53.43.43.15.1.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 53.43.43.15.1.2

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

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

Etapa 53.43.43.15.2

Some $1$ e $3$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin^{4} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.16

Multiplique o numerador pelo inverso do denominador.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.17

Combine.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.18

Multiplique $\cos^{3} (x)$ por $\cos (x)$ somando os expoentes.

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Etapa 53.43.43.18.1

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

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Etapa 53.43.43.18.1.1

Eleve $\cos (x)$ à potência de $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.18.1.2

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

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{{\cos (x)}^{3 + 1}} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.18.2

Some $3$ e $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.19

Multiplique $\sin^{4} (x)$ por $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.20

Converta de $\frac{\sin^{4} (x)}{\cos^{4} (x)}$ em $\tan^{4} (x)$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.21

Multiplique o numerador pelo inverso do denominador.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.22

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \sec (x) = \frac{0}{\cos (x)}$

Etapa 53.43.43.23

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos^{3} (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.24

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos (x) \cos^{2} (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.25

Separe as frações.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\sec (x)}{\cos (x)}) = \frac{0}{\cos (x)}$

Etapa 53.43.43.26

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\frac{1}{\cos (x)}}{\cos (x)}) = \frac{0}{\cos (x)}$

Etapa 53.43.43.27

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)})) = \frac{0}{\cos (x)}$

Etapa 53.43.43.28

Simplifique.

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Etapa 53.43.43.28.1

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) (\frac{1}{\cos (x)}))) = \frac{0}{\cos (x)}$

Etapa 53.43.43.28.2

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Etapa 53.43.43.28.3

Eleve $\sec (x)$ à potência de $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Etapa 53.43.43.28.4

Eleve $\sec (x)$ à potência de $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Etapa 53.43.43.28.5

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot {\sec (x)}^{1 + 1}) = \frac{0}{\cos (x)}$

Etapa 53.43.43.28.6

Some $1$ e $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 53.43.43.29

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1^{2}}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 53.43.43.30

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - ({(\frac{1}{\cos (x)})}^{2} \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 53.43.43.31

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\sec^{2} (x) \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 53.43.43.32

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

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Etapa 53.43.43.32.1

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - {\sec (x)}^{2 + 2} = \frac{0}{\cos (x)}$

Etapa 53.43.43.32.2

Some $2$ e $2$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{\cos (x)}$

Etapa 53.43.43.33

Separe as frações.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Etapa 53.43.43.34

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \sec (x)$

Etapa 53.43.43.35

Divida $0$ por $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0 \sec (x)$

Etapa 53.43.43.36

Multiplique $0$ por $\sec (x)$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 53.43.43.37

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

$1 + 2 (\sec^{2} (x) - 1) + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 53.43.43.38

Simplifique cada termo.

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Etapa 53.43.43.38.1

Aplique a propriedade distributiva.

$1 + 2 \sec^{2} (x) + 2 \cdot - 1 + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 53.43.43.38.2

Multiplique $2$ por $- 1$ .

$1 + 2 \sec^{2} (x) - 2 + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 53.43.43.39

Subtraia $2$ de $1$ .

$2 \sec^{2} (x) - 1 + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 53.43.43.40

Reordene o polinômio.

$\tan^{4} (x) - \sec^{4} (x) + 2 \sec^{2} (x) - 1 = 0$

Etapa 53.43.43.41

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

$(1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42

Simplifique.

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Etapa 53.43.43.42.1

Simplifique o lado esquerdo.

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Etapa 53.43.43.42.1.1

Simplifique $(1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x)) \cos (x)$ .

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Etapa 53.43.43.42.1.1.1

Simplifique cada termo.

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Etapa 53.43.43.42.1.1.1.1

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

$(1 + 2 {(\frac{\sin (x)}{\cos (x)})}^{2} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.1.2

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

$(1 + 2 \frac{\sin^{2} (x)}{\cos^{2} (x)} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.1.3

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.1.4

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + {(\frac{\sin (x)}{\cos (x)})}^{4} - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.1.5

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.1.6

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - {(\frac{1}{\cos (x)})}^{4}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.1.7

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

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \frac{1^{4}}{\cos^{4} (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.1.8

Um elevado a qualquer potência é um.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \frac{1}{\cos^{4} (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.2

Aplique a propriedade distributiva.

$1 \cos (x) + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.3

Simplifique.

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Etapa 53.43.43.42.1.1.3.1

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

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.3.2

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

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Etapa 53.43.43.42.1.1.3.2.1

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

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.3.2.2

Cancele o fator comum.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.3.2.3

Reescreva a expressão.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.3.3

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

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Etapa 53.43.43.42.1.1.3.3.1

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

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos (x) \cos^{3} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.3.3.2

Cancele o fator comum.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos (x) \cos^{3} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.3.3.3

Reescreva a expressão.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.3.4

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

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Etapa 53.43.43.42.1.1.3.4.1

Mova o negativo de maior ordem em $- \frac{1}{\cos^{4} (x)}$ para o numerador.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.3.4.2

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

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos (x) \cos^{3} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.3.4.3

Cancele o fator comum.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos (x) \cos^{3} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.3.4.4

Reescreva a expressão.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.4

Mova o número negativo para a frente da fração.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.5

Simplifique cada termo.

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Etapa 53.43.43.42.1.1.5.1

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

$\cos (x) + \frac{2 (\sin (x) \sin (x))}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.5.2

Separe as frações.

$\cos (x) + \frac{2 (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.5.3

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

$\cos (x) + \frac{2 (\sin (x))}{1} \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.5.4

Divida $2 (\sin (x))$ por $1$ .

$\cos (x) + 2 (\sin (x)) \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Etapa 53.43.43.42.1.1.5.5

Fatore $\sin^{3} (x)$ de $\sin^{4} (x)$ .

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

Etapa 53.43.43.42.1.1.5.6

Multiplique por $1$ .

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

Etapa 53.43.43.42.1.1.5.7

Separe as frações.

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

Etapa 53.43.43.42.1.1.5.8

Converta de $\frac{\sin^{3} (x)}{\cos^{3} (x)}$ em $\tan^{3} (x)$ .

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

Etapa 53.43.43.42.1.1.5.9

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

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

Etapa 53.43.43.42.2

Simplifique o lado direito.

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Etapa 53.43.43.42.2.1

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

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Etapa 53.43.43.42.2.1.1

Cancele o fator comum.

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

Etapa 53.43.43.42.2.1.2

Reescreva a expressão.

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

Etapa 53.43.43.43

Resolva $x$ .

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Etapa 53.43.43.43.1

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

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

Etapa 53.43.43.43.2

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

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Etapa 53.43.43.43.2.1

Cancele o fator comum.

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

Etapa 53.43.43.43.2.2

Reescreva a expressão.

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

Etapa 53.43.43.43.3

Simplifique o numerador.

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Etapa 53.43.43.43.3.1

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

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

Etapa 53.43.43.43.3.2

Combine expoentes.

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Etapa 53.43.43.43.3.2.1

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

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

Etapa 53.43.43.43.3.2.2

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

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

Etapa 53.43.43.43.3.2.3

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 53.43.43.43.3.2.4

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 53.43.43.43.3.2.5

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

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

Etapa 53.43.43.43.3.2.6

Some $1$ e $1$ .

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

Etapa 53.43.43.43.4

Multiplique o numerador pelo inverso do denominador.

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

Etapa 53.43.43.43.5

Combine frações.

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Etapa 53.43.43.43.5.1

Combine.

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

Etapa 53.43.43.43.5.2

Multiplique $2$ por $1$ .

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

Etapa 53.43.43.43.6

Simplifique o denominador.

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Etapa 53.43.43.43.6.1

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

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

Etapa 53.43.43.43.6.2

Some $1$ e $1$ .

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

Etapa 53.43.43.43.7

Multiplique por $1$ .

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

Etapa 53.43.43.43.8

Multiplique por $1$ .

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

Etapa 53.43.43.43.9

Separe as frações.

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

Etapa 53.43.43.43.10

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

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

Etapa 53.43.43.43.11

Multiplique $2$ por $1$ .

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

Etapa 53.43.43.43.12

Divida $2$ por $1$ .

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

Etapa 53.43.43.43.13

Simplifique o numerador.

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Etapa 53.43.43.43.13.1

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

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

Etapa 53.43.43.43.13.2

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

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

Etapa 53.43.43.43.14

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

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

Etapa 53.43.43.43.15

Multiplique $\sin (x)$ por $\sin^{3} (x)$ somando os expoentes.

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Etapa 53.43.43.43.15.1

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

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Etapa 53.43.43.43.15.1.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 53.43.43.43.15.1.2

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

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

Etapa 53.43.43.43.15.2

Some $1$ e $3$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin^{4} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.16

Multiplique o numerador pelo inverso do denominador.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.17

Combine.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.18

Multiplique $\cos^{3} (x)$ por $\cos (x)$ somando os expoentes.

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Etapa 53.43.43.43.18.1

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

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Etapa 53.43.43.43.18.1.1

Eleve $\cos (x)$ à potência de $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.18.1.2

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

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{{\cos (x)}^{3 + 1}} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.18.2

Some $3$ e $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.19

Multiplique $\sin^{4} (x)$ por $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.20

Converta de $\frac{\sin^{4} (x)}{\cos^{4} (x)}$ em $\tan^{4} (x)$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.21

Multiplique o numerador pelo inverso do denominador.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.22

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \sec (x) = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.23

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos^{3} (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.24

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos (x) \cos^{2} (x)} = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.25

Separe as frações.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\sec (x)}{\cos (x)}) = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.26

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\frac{1}{\cos (x)}}{\cos (x)}) = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.27

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)})) = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.28

Simplifique.

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Etapa 53.43.43.43.28.1

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) (\frac{1}{\cos (x)}))) = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.28.2

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.28.3

Eleve $\sec (x)$ à potência de $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.28.4

Eleve $\sec (x)$ à potência de $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.28.5

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot {\sec (x)}^{1 + 1}) = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.28.6

Some $1$ e $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.29

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1^{2}}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.30

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - ({(\frac{1}{\cos (x)})}^{2} \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.31

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\sec^{2} (x) \sec^{2} (x)) = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.32

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

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Etapa 53.43.43.43.32.1

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - {\sec (x)}^{2 + 2} = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.32.2

Some $2$ e $2$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{\cos (x)}$

Etapa 53.43.43.43.33

Separe as frações.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Etapa 53.43.43.43.34

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

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \sec (x)$

Etapa 53.43.43.43.35

Divida $0$ por $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0 \sec (x)$

Etapa 53.43.43.43.36

Multiplique $0$ por $\sec (x)$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 53.43.43.43.37

Simplifique o lado esquerdo.

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Etapa 53.43.43.43.37.1

Simplifique $1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x)$ .

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Etapa 53.43.43.43.37.1.1

Fatore usando a regra do quadrado perfeito.

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Etapa 53.43.43.43.37.1.1.1

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

$1^{2} + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0$

Etapa 53.43.43.43.37.1.1.2

Reescreva $\tan^{4} (x)$ como ${(\tan^{2} (x))}^{2}$ .

$1^{2} + 2 \tan^{2} (x) + {(\tan^{2} (x))}^{2} - \sec^{4} (x) = 0$

Etapa 53.43.43.43.37.1.1.3

Verifique se o termo do meio é duas vezes o produto dos números ao quadrado no primeiro e no terceiro termos.

$2 \tan^{2} (x) = 2 \cdot 1 \cdot \tan^{2} (x)$

Etapa 53.43.43.43.37.1.1.4

Reescreva o polinômio.

$1^{2} + 2 \cdot 1 \cdot \tan^{2} (x) + {(\tan^{2} (x))}^{2} - \sec^{4} (x) = 0$

Etapa 53.43.43.43.37.1.1.5

Fatore usando a regra do trinômio quadrado perfeito $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , em que $a = 1$ e $b = \tan^{2} (x)$ .

${(1 + \tan^{2} (x))}^{2} - \sec^{4} (x) = 0$

Etapa 53.43.43.43.37.1.2

Reescreva $\sec^{4} (x)$ como ${(\sec^{2} (x))}^{2}$ .

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

Etapa 53.43.43.43.37.1.3

Como os dois termos são quadrados perfeitos, fatore usando a fórmula da diferença de quadrados, $a^{2} - b^{2} = (a + b) (a - b)$ em que $a = 1 + \tan^{2} (x)$ e $b = \sec^{2} (x)$ .

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

Etapa 53.43.43.43.37.1.4

Reorganize os termos.

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

Etapa 53.43.43.43.37.1.5

Aplique a identidade trigonométrica fundamental.

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

Etapa 53.43.43.43.37.1.6

Some $\sec^{2} (x)$ e $\sec^{2} (x)$ .

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

Etapa 53.43.43.43.37.1.7

Reorganize os termos.

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

Etapa 53.43.43.43.37.1.8

Aplique a identidade trigonométrica fundamental.

$2 \sec^{2} (x) (\sec^{2} (x) - \sec^{2} (x)) = 0$

Etapa 53.43.43.43.37.1.9

Subtraia $\sec^{2} (x)$ de $\sec^{2} (x)$ .

$2 \sec^{2} (x) \cdot 0 = 0$

Etapa 53.43.43.43.37.1.10

Multiplique $2 \sec^{2} (x) \cdot 0$ .

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Etapa 53.43.43.43.37.1.10.1

Multiplique $0$ por $2$ .

$0 \sec^{2} (x) = 0$

Etapa 53.43.43.43.37.1.10.2

Multiplique $0$ por $\sec^{2} (x)$ .

$0 = 0$

Etapa 53.43.43.43.38

Como $0 = 0$ , a equação sempre será verdadeira para qualquer valor de $x$ .

Todos os números reais

Etapa 54

O resultado pode ser mostrado de várias formas.

Todos os números reais

Notação de intervalo:

$(- \infty, \infty)$