Trigonometria Exemplos

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

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

Etapa 1

Comece do lado direito.

{(cot (x) - csc (x))}^{2}

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

Etapa 2

Converta em senos e cossenos.

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Etapa 2.1

Escreva

cot (x)

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

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

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

Etapa 2.2

Aplique a identidade recíproca a

csc (x)

$\csc (x)$ .

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

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

Etapa 2.3

Simplifique.

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Etapa 2.3.1

Reescreva

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

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

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

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

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

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

Etapa 2.3.2

Expanda

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

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

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Etapa 2.3.2.1

Aplique a propriedade distributiva.

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

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

Etapa 2.3.2.2

Aplique a propriedade distributiva.

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

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

Etapa 2.3.2.3

Aplique a propriedade distributiva.

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

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

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

Etapa 2.3.3

Simplifique e combine termos semelhantes.

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Etapa 2.3.3.1

Simplifique cada termo.

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Etapa 2.3.3.1.1

Multiplique

\frac{cos (x)}{sin (x)} \cdot \frac{cos (x)}{sin (x)}

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

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Etapa 2.3.3.1.1.1

Multiplique

\frac{cos (x)}{sin (x)}

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

\frac{cos (x)}{sin (x)}

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

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

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

Etapa 2.3.3.1.1.2

Eleve

cos (x)

$\cos (x)$ à potência de

1

$1$ .

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

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

Etapa 2.3.3.1.1.3

Eleve

cos (x)

$\cos (x)$ à potência de

1

$1$ .

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

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

Etapa 2.3.3.1.1.4

Use a regra da multiplicação de potências

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

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

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

Etapa 2.3.3.1.1.5

Some

1

$1$ e

1

$1$ .

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

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

Etapa 2.3.3.1.1.6

Eleve

sin (x)

$\sin (x)$ à potência de

1

$1$ .

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

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

Etapa 2.3.3.1.1.7

Eleve

sin (x)

$\sin (x)$ à potência de

1

$1$ .

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

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

Etapa 2.3.3.1.1.8

Use a regra da multiplicação de potências

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

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

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

Etapa 2.3.3.1.1.9

Some

$1$ e

$1$ .

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

Etapa 2.3.3.1.2

Multiplique

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

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Etapa 2.3.3.1.2.1

Multiplique

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

$\frac{1}{\sin (x)}$ .

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

Etapa 2.3.3.1.2.2

Eleve

$\sin (x)$ à potência de

$1$ .

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

Etapa 2.3.3.1.2.3

Eleve

$\sin (x)$ à potência de

$1$ .

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

Etapa 2.3.3.1.2.4

Use a regra da multiplicação de potências

$a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

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

Etapa 2.3.3.1.2.5

Some

$1$ e

$1$ .

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

Etapa 2.3.3.1.3

Multiplique

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

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Etapa 2.3.3.1.3.1

Multiplique

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

$\frac{1}{\sin (x)}$ .

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

Etapa 2.3.3.1.3.2

Eleve

$\sin (x)$ à potência de

$1$ .

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

Etapa 2.3.3.1.3.3

Eleve

$\sin (x)$ à potência de

$1$ .

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

Etapa 2.3.3.1.3.4

Use a regra da multiplicação de potências

$a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

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

Etapa 2.3.3.1.3.5

Some

$1$ e

$1$ .

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

Etapa 2.3.3.1.4

Multiplique

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

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Etapa 2.3.3.1.4.1

Multiplique

$- 1$ por

$- 1$ .

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

Etapa 2.3.3.1.4.2

Multiplique

$\frac{1}{\sin (x)}$ por

$1$ .

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

Etapa 2.3.3.1.4.3

Multiplique

$\frac{1}{\sin (x)}$ por

$\frac{1}{\sin (x)}$ .

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

Etapa 2.3.3.1.4.4

Eleve

$\sin (x)$ à potência de

$1$ .

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

Etapa 2.3.3.1.4.5

Eleve

$\sin (x)$ à potência de

$1$ .

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

Etapa 2.3.3.1.4.6

Use a regra da multiplicação de potências

$a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

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

Etapa 2.3.3.1.4.7

Some

$1$ e

$1$ .

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

Etapa 2.3.3.2

Combine os numeradores em relação ao denominador comum.

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

Etapa 2.3.4

Combine os numeradores em relação ao denominador comum.

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

Etapa 2.3.5

Subtraia

$\cos (x)$ de

$- \cos (x)$ .

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

Etapa 2.3.6

Fatore usando a regra do quadrado perfeito.

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

Etapa 2.4

Simplifique.

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Etapa 2.4.1

Reescreva

${(\cos (x) - 1)}^{2}$ como

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

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

Etapa 2.4.2

Expanda

$(\cos (x) - 1) (\cos (x) - 1)$ usando o método FOIL.

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Etapa 2.4.2.1

Aplique a propriedade distributiva.

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

Etapa 2.4.2.2

Aplique a propriedade distributiva.

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

Etapa 2.4.2.3

Aplique a propriedade distributiva.

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

Etapa 2.4.3

Simplifique e combine termos semelhantes.

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

Etapa 3

Fatore usando a regra do quadrado perfeito.

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

Etapa 4

Aplique a identidade trigonométrica fundamental inversa.

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

Etapa 5

Simplifique.

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Etapa 5.1

Simplifique o denominador.

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Etapa 5.1.1

Reescreva

$1$ como

$1^{2}$ .

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

Etapa 5.1.2

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

$b = \cos (x)$ .

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

Etapa 5.2

Cancele o fator comum de

${(\cos (x) - 1)}^{2}$ e

$1 - \cos (x)$ .

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

Etapa 6

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

$\frac{1 - \cos (x)}{1 + \cos (x)} = {(\cot (x) - \csc (x))}^{2}$ é uma identidade