Trigonometria Exemplos

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

Etapa 1

Simplifique cada termo.

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Etapa 1.1

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

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

Etapa 1.2

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

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

Etapa 2

Para escrever $\frac{\cos (x)}{1 - \frac{\sin (x)}{\cos (x)}}$ como fração com um denominador comum, multiplique por $\frac{1 - \frac{\cos (x)}{\sin (x)}}{1 - \frac{\cos (x)}{\sin (x)}}$ .

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

Etapa 3

Para escrever $\frac{\sin (x)}{1 - \frac{\cos (x)}{\sin (x)}}$ como fração com um denominador comum, multiplique por $\frac{1 - \frac{\sin (x)}{\cos (x)}}{1 - \frac{\sin (x)}{\cos (x)}}$ .

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

Etapa 4

Escreva cada expressão com um denominador comum de $(1 - \frac{\sin (x)}{\cos (x)}) (1 - \frac{\cos (x)}{\sin (x)})$ , multiplicando cada um por um fator apropriado de $1$ .

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Etapa 4.1

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

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

Etapa 4.2

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

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

Etapa 4.3

Reordene os fatores de $(1 - \frac{\sin (x)}{\cos (x)}) (1 - \frac{\cos (x)}{\sin (x)})$ .

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

Etapa 5

Combine os numeradores em relação ao denominador comum.

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

Etapa 6

Simplifique o numerador.

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Etapa 6.1

Aplique a propriedade distributiva.

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

Etapa 6.2

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

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

Etapa 6.3

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 6.4

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

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Etapa 6.4.1

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

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

Etapa 6.4.2

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 6.4.3

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 6.4.4

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

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

Etapa 6.4.5

Some $1$ e $1$ .

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

Etapa 6.5

Aplique a propriedade distributiva.

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

Etapa 6.6

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

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

Etapa 6.7

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 6.8

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

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Etapa 6.8.1

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

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

Etapa 6.8.2

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 6.8.3

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 6.8.4

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

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

Etapa 6.8.5

Some $1$ e $1$ .

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

Etapa 7

Simplifique o numerador.

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Etapa 7.1

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

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

Etapa 7.2

Separe as frações.

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

Etapa 7.3

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

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

Etapa 7.4

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

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

Etapa 7.5

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

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

Etapa 7.6

Separe as frações.

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

Etapa 7.7

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

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

Etapa 7.8

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

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

Etapa 8

Simplifique o denominador.

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Etapa 8.1

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

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

Etapa 8.2

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

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