Trigonometria Exemplos

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

Etapa 1

Comece do lado esquerdo.

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

Etapa 2

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

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

Etapa 3

Combine.

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

Etapa 4

Simplifique o numerador.

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Etapa 4.1

Expanda $(\tan (x) + \sec (x) - 1) (- \tan (x) + \sec (x) + 1)$ multiplicando cada termo na primeira expressão por cada um dos termos na segunda expressão.

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

Etapa 4.2

Simplifique cada termo.

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Etapa 4.2.1

Multiplique $\tan (x) (- \tan (x))$ .

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Etapa 4.2.1.1

Eleve $\tan (x)$ à potência de $1$ .

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

Etapa 4.2.1.2

Eleve $\tan (x)$ à potência de $1$ .

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

Etapa 4.2.1.3

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

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

Etapa 4.2.1.4

Some $1$ e $1$ .

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

Etapa 4.2.2

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

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

Etapa 4.2.3

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

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Etapa 4.2.3.1

Eleve $\sec (x)$ à potência de $1$ .

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

Etapa 4.2.3.2

Eleve $\sec (x)$ à potência de $1$ .

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

Etapa 4.2.3.3

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

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

Etapa 4.2.3.4

Some $1$ e $1$ .

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

Etapa 4.2.4

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

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

Etapa 4.2.5

Multiplique $- 1 (- \tan (x))$ .

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Etapa 4.2.5.1

Multiplique $- 1$ por $- 1$ .

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

Etapa 4.2.5.2

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

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

Etapa 4.2.6

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

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

Etapa 4.2.7

Multiplique $- 1$ por $1$ .

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

Etapa 4.3

Some $\sec (x) \tan (x)$ e $\sec (x) (- \tan (x))$ .

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Etapa 4.3.1

Reordene $\sec (x)$ e $- 1$ .

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

Etapa 4.3.2

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

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

Etapa 4.4

Some $- {\tan (x)}^{2}$ e $0$ .

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

Etapa 4.5

Some $\tan (x)$ e $\tan (x)$ .

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

Etapa 4.6

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

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

Etapa 4.7

Some $- {\tan (x)}^{2}$ e $0$ .

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

Etapa 5

Simplifique o denominador.

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Etapa 5.1

Expanda $(\tan (x) - \sec (x) + 1) (- \tan (x) + \sec (x) + 1)$ multiplicando cada termo na primeira expressão por cada um dos termos na segunda expressão.

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

Etapa 5.2

Simplifique cada termo.

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Etapa 5.2.1

Multiplique $\tan (x) (- \tan (x))$ .

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Etapa 5.2.1.1

Eleve $\tan (x)$ à potência de $1$ .

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

Etapa 5.2.1.2

Eleve $\tan (x)$ à potência de $1$ .

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

Etapa 5.2.1.3

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

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

Etapa 5.2.1.4

Some $1$ e $1$ .

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

Etapa 5.2.2

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

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

Etapa 5.2.3

Multiplique $- \sec (x) (- \tan (x))$ .

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Etapa 5.2.3.1

Multiplique $- 1$ por $- 1$ .

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

Etapa 5.2.3.2

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

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

Etapa 5.2.4

Multiplique $- \sec (x) \sec (x)$ .

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Etapa 5.2.4.1

Eleve $\sec (x)$ à potência de $1$ .

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

Etapa 5.2.4.2

Eleve $\sec (x)$ à potência de $1$ .

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

Etapa 5.2.4.3

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

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

Etapa 5.2.4.4

Some $1$ e $1$ .

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

Etapa 5.2.5

Multiplique $- 1$ por $1$ .

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

Etapa 5.2.6

Multiplique $- \tan (x)$ por $1$ .

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

Etapa 5.2.7

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

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

Etapa 5.2.8

Multiplique $1$ por $1$ .

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

Etapa 5.3

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

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

Etapa 5.4

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

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

Etapa 5.5

Some $- {\tan (x)}^{2}$ e $0$ .

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

Etapa 5.6

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

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

Etapa 5.7

Some $- {\tan (x)}^{2}$ e $0$ .

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

Etapa 6

Simplifique a expressão.

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Etapa 6.1

Mova $\sec^{2} (x)$ .

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

Etapa 6.2

Reordene $- \tan^{2} (x)$ e $\sec^{2} (x)$ .

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

Etapa 6.3

Aplique a identidade trigonométrica fundamental.

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

Etapa 6.4

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

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

Etapa 6.5

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

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

Etapa 6.6

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

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

Etapa 6.7

Aplique a identidade trigonométrica fundamental.

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

Etapa 6.8

Simplifique o numerador.

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Etapa 6.8.1

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

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

Etapa 6.8.2

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

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

Etapa 6.8.3

Subtraia $1$ de $1$ .

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

Etapa 6.8.4

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

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

Etapa 6.9

Simplifique o denominador.

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Etapa 6.9.1

Subtraia $\tan^{2} (x)$ de $- \tan^{2} (x)$ .

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

Etapa 6.9.2

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

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Etapa 6.9.2.1

Fatore $2 \tan (x)$ de $2 \sec (x) \tan (x)$ .

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

Etapa 6.9.2.2

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

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

Etapa 6.9.2.3

Fatore $2 \tan (x)$ de $2 \tan (x) (\sec (x)) + 2 \tan (x) (- \tan (x))$ .

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

Etapa 6.9.3

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

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

Etapa 6.9.4

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

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

Etapa 6.9.5

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

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

Etapa 6.9.6

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

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

Etapa 6.10

Cancele o fator comum de $\frac{2 \sin (x)}{\cos (x)}$ .

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Etapa 6.10.1

Cancele o fator comum.

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

Etapa 6.10.2

Reescreva a expressão.

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

Etapa 7

Simplifique.

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Etapa 7.1

Combine os numeradores em relação ao denominador comum.

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

Etapa 7.2

Multiplique o numerador pelo inverso do denominador.

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

Etapa 7.3

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

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

Etapa 8

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

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

Etapa 9

Combine.

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

Etapa 10

Simplifique o numerador.

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Etapa 10.1

Aplique a propriedade distributiva.

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

Etapa 10.2

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

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

Etapa 11

Simplifique o denominador.

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Etapa 11.1

Expanda $(1 - \sin (x)) (1 + \sin (x))$ usando o método FOIL.

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Etapa 11.1.1

Aplique a propriedade distributiva.

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

Etapa 11.1.2

Aplique a propriedade distributiva.

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

Etapa 11.1.3

Aplique a propriedade distributiva.

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

Etapa 11.2

Simplifique e combine termos semelhantes.

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

Etapa 12

Aplique a identidade trigonométrica fundamental.

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

Etapa 13

Simplifique.

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Etapa 13.1

Fatore $\cos (x)$ de $\cos (x) + \cos (x) \sin (x)$ .

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Etapa 13.1.1

Multiplique por $1$ .

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

Etapa 13.1.2

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

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

Etapa 13.1.3

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

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

Etapa 13.2

Cancele os fatores comuns.

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

Etapa 14

Agora, considere o lado direito da equação.

$\tan (x) + \sec (x)$

Etapa 15

Converta em senos e cossenos.

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Etapa 15.1

Escreva $\tan (x)$ nos senos e cossenos usando a fórmula do quociente.

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

Etapa 15.2

Aplique a identidade recíproca a $\sec (x)$ .

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

Etapa 16

Combine os numeradores em relação ao denominador comum.

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

Etapa 17

Reordene os termos.

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

Etapa 18

Como os dois lados demonstraram ser equivalentes, a equação é uma identidade.

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