Álgebra Exemplos

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

Etapa 1

Simplifique com fatoração.

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Etapa 1.1

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

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

Etapa 1.2

Fatore $\cot (x)$ de $- \cot (x)$ .

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

Etapa 1.3

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

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

Etapa 2

Aplique a identidade trigonométrica fundamental.

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

Etapa 3

Simplifique o numerador.

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Etapa 3.1

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

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

Etapa 3.2

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

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

Etapa 3.3

Aplique a regra do produto a $\frac{\sin (x)}{\cos (x)}$ .

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

Etapa 4

Simplifique o denominador.

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Etapa 4.1

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

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

Etapa 4.2

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

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Etapa 4.2.1

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

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

Etapa 4.2.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 4.2.3

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 4.2.4

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

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

Etapa 4.2.5

Some $1$ e $1$ .

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

Etapa 5

Multiplique $\frac{\cos (x)}{\sin (x)}$ por $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ .

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

Etapa 6

Simplifique o numerador.

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Etapa 6.1

Reduza a expressão $\frac{\cos (x) \sin^{2} (x)}{\sin (x) \cos^{2} (x)}$ cancelando os fatores comuns.

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Etapa 6.1.1

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

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

Etapa 6.1.2

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

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

Etapa 6.1.3

Cancele o fator comum.

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

Etapa 6.1.4

Reescreva a expressão.

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

Etapa 6.2

Cancele o fator comum de $\sin^{2} (x)$ e $\sin (x)$ .

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Etapa 6.2.1

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

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

Etapa 6.2.2

Cancele os fatores comuns.

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Etapa 6.2.2.1

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

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

Etapa 6.2.2.2

Cancele o fator comum.

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

Etapa 6.2.2.3

Reescreva a expressão.

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

Etapa 7

Multiplique o numerador pelo inverso do denominador.

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

Etapa 8

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

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

Etapa 9

Separe as frações.

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

Etapa 10

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

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

Etapa 11

Simplifique cada termo.

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Etapa 11.1

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

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

Etapa 11.2

Separe as frações.

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

Etapa 11.3

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

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

Etapa 11.4

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

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

Etapa 12

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

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