Trigonometria Exemplos

\tan (x) (\sin (x) + \cot (x) \cdot \cos (x))

$\tan (x) (\sin (x) + \cot (x) \cdot \cos (x))$

Etapa 1

Reescreva

\tan (x)

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

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

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

Etapa 2

Simplifique cada termo.

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Etapa 2.1

Reescreva

\cot (x)

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

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

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

Etapa 2.2

Multiplique

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

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

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Etapa 2.2.1

Combine

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

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

\cos (x)

$\cos (x)$ .

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

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

Etapa 2.2.2

Eleve

\cos (x)

$\cos (x)$ à potência de

1

$1$ .

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

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

Etapa 2.2.3

Eleve

\cos (x)

$\cos (x)$ à potência de

1

$1$ .

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

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

Etapa 2.2.4

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

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

Etapa 2.2.5

Some

1

$1$ e

1

$1$ .

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

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

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

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

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

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

Etapa 3

Aplique a propriedade distributiva.

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

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

Etapa 4

Multiplique

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

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

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Etapa 4.1

Combine

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

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

\sin (x)

$\sin (x)$ .

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

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

Etapa 4.2

Eleve

\sin (x)

$\sin (x)$ à potência de

1

$1$ .

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

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

Etapa 4.3

Eleve

\sin (x)

$\sin (x)$ à potência de

1

$1$ .

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

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

Etapa 4.4

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

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

Etapa 4.5

Some

1

$1$ e

1

$1$ .

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

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

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

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

Etapa 5

Combine.

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

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

Etapa 6

Simplifique cada termo.

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Etapa 6.1

Cancele o fator comum de

\sin (x)

$\sin (x)$ .

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Etapa 6.1.1

Cancele o fator comum.

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

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

Etapa 6.1.2

Reescreva a expressão.

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

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

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

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

Etapa 6.2

Cancele o fator comum de

\cos^{2} (x)

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

\cos (x)

$\cos (x)$ .

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Etapa 6.2.1

Fatore

\cos (x)

$\cos (x)$ de

\cos^{2} (x)

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

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

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

Etapa 6.2.2

Cancele os fatores comuns.

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Etapa 6.2.2.1

Multiplique por

1

$1$ .

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

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

Etapa 6.2.2.2

Cancele o fator comum.

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

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

Etapa 6.2.2.3

Reescreva a expressão.

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

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

Etapa 6.2.2.4

Divida

\cos (x)

$\cos (x)$ por

1

$1$ .

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

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

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

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

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

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

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

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

Etapa 7

Simplifique cada termo.

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Etapa 7.1

Fatore

\sin (x)

$\sin (x)$ de

\sin^{2} (x)

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

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

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

Etapa 7.2

Separe as frações.

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

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

Etapa 7.3

Converta de

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

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

\tan (x)

$\tan (x)$ .

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

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

Etapa 7.4

Divida

\sin (x)

$\sin (x)$ por

1

$1$ .

\sin (x) \tan (x) + \cos (x)

$\sin (x) \tan (x) + \cos (x)$

\sin (x) \tan (x) + \cos (x)

$\sin (x) \tan (x) + \cos (x)$

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