Pré-cálculo Exemplos

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

Etapa 1

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

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

Etapa 2

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

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

Etapa 3

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

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Etapa 3.1

Aplique a propriedade distributiva.

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

Etapa 3.2

Aplique a propriedade distributiva.

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

Etapa 3.3

Aplique a propriedade distributiva.

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

Etapa 4

Simplifique e combine termos semelhantes.

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Etapa 4.1

Simplifique cada termo.

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Etapa 4.1.1

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

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Etapa 4.1.1.1

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

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

Etapa 4.1.1.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 4.1.1.3

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 4.1.1.4

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

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

Etapa 4.1.1.5

Some $1$ e $1$ .

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

Etapa 4.1.1.6

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 4.1.1.7

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 4.1.1.8

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

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

Etapa 4.1.1.9

Some $1$ e $1$ .

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

Etapa 4.1.2

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

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

Etapa 4.1.3

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

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

Etapa 4.1.4

Multiplique $1$ por $1$ .

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

Etapa 4.2

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

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

Etapa 5

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

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

Etapa 6

Aplique a propriedade distributiva.

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

Etapa 7

Simplifique.

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Etapa 7.1

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

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Etapa 7.1.1

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

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

Etapa 7.1.2

Cancele o fator comum.

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

Etapa 7.1.3

Reescreva a expressão.

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

Etapa 7.2

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

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Etapa 7.2.1

Cancele o fator comum.

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

Etapa 7.2.2

Reescreva a expressão.

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

Etapa 7.3

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

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

Etapa 8

Simplifique cada termo.

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Etapa 8.1

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

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

Etapa 8.2

Separe as frações.

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

Etapa 8.3

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

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

Etapa 8.4

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

$\sin (x) \tan (x) + 2 \sin (x) + \cos (x)$