Trigonometria Exemplos

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

Etapa 1

Subtraia $\frac{\sin (x)}{\sin (x) + \cos (x)}$ dos dois lados da equação.

$\frac{\tan (x)}{1 + \tan (x)} - \frac{\sin (x)}{\sin (x) + \cos (x)} = 0$

Etapa 2

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

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Etapa 2.1

Simplifique cada termo.

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Etapa 2.1.1

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

$\frac{\frac{\sin (x)}{\cos (x)}}{1 + \tan (x)} - \frac{\sin (x)}{\sin (x) + \cos (x)} = 0$

Etapa 2.1.2

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

$\frac{\frac{\sin (x)}{\cos (x)}}{1 + \frac{\sin (x)}{\cos (x)}} - \frac{\sin (x)}{\sin (x) + \cos (x)} = 0$

Etapa 2.1.3

Multiplique o numerador pelo inverso do denominador.

$\frac{\sin (x)}{\cos (x)} \cdot \frac{1}{1 + \frac{\sin (x)}{\cos (x)}} - \frac{\sin (x)}{\sin (x) + \cos (x)} = 0$

Etapa 2.1.4

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

$\frac{\sin (x)}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})} - \frac{\sin (x)}{\sin (x) + \cos (x)} = 0$

Etapa 2.2

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

$\frac{\sin (x)}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})} \cdot \frac{\sin (x) + \cos (x)}{\sin (x) + \cos (x)} - \frac{\sin (x)}{\sin (x) + \cos (x)} = 0$

Etapa 2.3

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

$\frac{\sin (x)}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})} \cdot \frac{\sin (x) + \cos (x)}{\sin (x) + \cos (x)} - \frac{\sin (x)}{\sin (x) + \cos (x)} \cdot \frac{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})} = 0$

Etapa 2.4

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

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Etapa 2.4.1

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

$\frac{\sin (x) (\sin (x) + \cos (x))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} - \frac{\sin (x)}{\sin (x) + \cos (x)} \cdot \frac{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})} = 0$

Etapa 2.4.2

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

$\frac{\sin (x) (\sin (x) + \cos (x))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} - \frac{\sin (x) (\cos (x) (1 + \frac{\sin (x)}{\cos (x)}))}{(\sin (x) + \cos (x)) (\cos (x) (1 + \frac{\sin (x)}{\cos (x)}))} = 0$

Etapa 2.4.3

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

$\frac{\sin (x) (\sin (x) + \cos (x))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} - \frac{\sin (x) (\cos (x) (1 + \frac{\sin (x)}{\cos (x)}))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Etapa 2.5

Combine os numeradores em relação ao denominador comum.

$\frac{\sin (x) (\sin (x) + \cos (x)) - \sin (x) (\cos (x) (1 + \frac{\sin (x)}{\cos (x)}))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Etapa 2.6

Simplifique o numerador.

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Etapa 2.6.1

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

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Etapa 2.6.1.1

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

$\frac{\sin (x) (\sin (x) + \cos (x)) + \sin (x) (- \cos (x) (1 + \frac{\sin (x)}{\cos (x)}))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Etapa 2.6.1.2

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

$\frac{\sin (x) (\sin (x) + \cos (x) - \cos (x) (1 + \frac{\sin (x)}{\cos (x)}))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Etapa 2.6.2

Aplique a propriedade distributiva.

$\frac{\sin (x) (\sin (x) + \cos (x) - \cos (x) \cdot 1 - \cos (x) \frac{\sin (x)}{\cos (x)})}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Etapa 2.6.3

Multiplique $- 1$ por $1$ .

$\frac{\sin (x) (\sin (x) + \cos (x) - \cos (x) - \cos (x) \frac{\sin (x)}{\cos (x)})}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Etapa 2.6.4

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

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Etapa 2.6.4.1

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

$\frac{\sin (x) (\sin (x) + \cos (x) - \cos (x) + \cos (x) \cdot - 1 \frac{\sin (x)}{\cos (x)})}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Etapa 2.6.4.2

Cancele o fator comum.

$\frac{\sin (x) (\sin (x) + \cos (x) - \cos (x) + \cos (x) \cdot - 1 \frac{\sin (x)}{\cos (x)})}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Etapa 2.6.4.3

Reescreva a expressão.

$\frac{\sin (x) (\sin (x) + \cos (x) - \cos (x) - \sin (x))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Etapa 2.6.5

Subtraia $\sin (x)$ de $\sin (x)$ .

$\frac{\sin (x) (0 + \cos (x) - \cos (x))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Etapa 2.6.6

Some $0$ e $\cos (x)$ .

$\frac{\sin (x) (\cos (x) - \cos (x))}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Etapa 2.6.7

Subtraia $\cos (x)$ de $\cos (x)$ .

$\frac{\sin (x) \cdot 0}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Etapa 2.7

Multiplique $\sin (x)$ por $0$ .

$\frac{0}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))} = 0$

Etapa 2.8

Divida $0$ por $\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))$ .

$0 = 0$

Etapa 3

O domínio da expressão consiste em todos os números reais, exceto quando a expressão é indefinida. Nesse caso, não existe um número real que torne a expressão indefinida.