Pré-cálculo Exemplos

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

Etapa 1

Comece do lado esquerdo.

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

Etapa 2

Converta em senos e cossenos.

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Etapa 2.1

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

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

Etapa 2.2

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

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

Etapa 3

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

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

Etapa 4

Combine.

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

Etapa 5

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

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

Etapa 6

Simplifique o denominador.

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Etapa 6.1

Expanda $(\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}) (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)})$ usando o método FOIL.

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Etapa 6.1.1

Aplique a propriedade distributiva.

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

Etapa 6.1.2

Aplique a propriedade distributiva.

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

Etapa 6.1.3

Aplique a propriedade distributiva.

Etapa 6.2

Simplifique e combine termos semelhantes.

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Etapa 6.2.1

Simplifique cada termo.

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Etapa 6.2.1.1

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

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Etapa 6.2.1.1.1

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

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

Etapa 6.2.1.1.2

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 6.2.1.1.3

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 6.2.1.1.4

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

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

Etapa 6.2.1.1.5

Some $1$ e $1$ .

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

Etapa 6.2.1.2

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

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Etapa 6.2.1.2.1

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

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

Etapa 6.2.1.2.2

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 6.2.1.2.3

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 6.2.1.2.4

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

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

Etapa 6.2.1.2.5

Some $1$ e $1$ .

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

Etapa 6.2.1.3

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

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Etapa 6.2.1.3.1

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

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

Etapa 6.2.1.3.2

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 6.2.1.3.3

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 6.2.1.3.4

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

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

Etapa 6.2.1.3.5

Some $1$ e $1$ .

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

Etapa 6.2.1.4

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

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Etapa 6.2.1.4.1

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

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

Etapa 6.2.1.4.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 6.2.1.4.3

Eleve $\sin (x)$ à potência de $1$ .

Etapa 6.2.1.4.4

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

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

Etapa 6.2.1.4.5

Some $1$ e $1$ .

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

Etapa 6.2.1.4.6

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 6.2.1.4.7

Eleve $\cos (x)$ à potência de $1$ .

Etapa 6.2.1.4.8

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

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

Etapa 6.2.1.4.9

Some $1$ e $1$ .

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

Etapa 6.2.2

Combine os numeradores em relação ao denominador comum.

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

Etapa 6.3

Combine os numeradores em relação ao denominador comum.

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

Etapa 6.4

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

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

Etapa 6.5

Fatore usando a regra do quadrado perfeito.

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

Etapa 7

Aplique a identidade trigonométrica fundamental inversa.

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

Etapa 8

Simplifique.

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Etapa 8.1

Multiplique o numerador pelo inverso do denominador.

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

Etapa 8.2

Simplifique o numerador.

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Etapa 8.2.1

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

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

Etapa 8.2.2

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 $a = 1$ e $b = \sin (x)$ .

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

Etapa 8.3

Cancele os fatores comuns.

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Etapa 8.3.1

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

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

Etapa 8.3.2

Cancele o fator comum.

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

Etapa 8.3.3

Reescreva a expressão.

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

Etapa 8.4

Aplique a propriedade distributiva.

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

Etapa 8.5

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

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

Etapa 8.6

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

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

Etapa 8.7

Combine os numeradores em relação ao denominador comum.

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

Etapa 8.8

Simplifique cada termo.

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Etapa 8.8.1

Aplique a propriedade distributiva.

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

Etapa 8.8.2

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

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

Etapa 8.8.3

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

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Etapa 8.8.3.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 8.8.3.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 8.8.3.3

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

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

Etapa 8.8.3.4

Some $1$ e $1$ .

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

Etapa 8.9

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

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

Etapa 8.10

Some $1$ e $0$ .

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

Etapa 8.11

Simplifique o numerador.

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Etapa 8.11.1

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

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

Etapa 8.11.2

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 $a = 1$ e $b = \sin (x)$ .

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

Etapa 8.12

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

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

Etapa 9

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

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