Trigonometría Ejemplos

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

Paso 1

Resta $\frac{\sin (x)}{\sin (x) + \cos (x)}$ de ambos lados de la ecuación.

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

Paso 2

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

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Paso 2.1

Simplifica cada término.

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Paso 2.1.1

Reescribe $\tan (x)$ en términos de senos y cosenos.

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

Paso 2.1.2

Reescribe $\tan (x)$ en términos de senos y cosenos.

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

Paso 2.1.3

Multiplica el numerador por la recíproca del denominador.

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

Paso 2.1.4

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

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

Paso 2.2

Para escribir $\frac{\sin (x)}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})}$ como una fracción con un denominador común, multiplica por $\frac{\sin (x) + \cos (x)}{\sin (x) + \cos (x)}$ .

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

Paso 2.3

Para escribir $- \frac{\sin (x)}{\sin (x) + \cos (x)}$ como una fracción con un denominador común, multiplica por $\frac{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})}{\cos (x) (1 + \frac{\sin (x)}{\cos (x)})}$ .

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

Paso 2.4

Escribe cada expresión con un denominador común de $\cos (x) (1 + \frac{\sin (x)}{\cos (x)}) (\sin (x) + \cos (x))$ , mediante la multiplicación de cada uno por un factor adecuado de $1$ .

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Paso 2.4.1

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

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

Paso 2.4.2

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

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

Paso 2.4.3

Reordena los factores de $(\sin (x) + \cos (x)) (\cos (x) (1 + \frac{\sin (x)}{\cos (x)}))$ .

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

Paso 2.5

Combina los numeradores sobre el denominador común.

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

Paso 2.6

Simplifica el numerador.

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Paso 2.6.1

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

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Paso 2.6.1.1

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

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

Paso 2.6.1.2

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

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

Paso 2.6.2

Aplica la propiedad distributiva.

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

Paso 2.6.3

Multiplica $- 1$ por $1$ .

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

Paso 2.6.4

Cancela el factor común de $\cos (x)$ .

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Paso 2.6.4.1

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

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

Paso 2.6.4.2

Cancela el factor común.

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

Paso 2.6.4.3

Reescribe la expresión.

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

Paso 2.6.5

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

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

Paso 2.6.6

Suma $0$ y $\cos (x)$ .

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

Paso 2.6.7

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

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

Paso 2.7

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

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

Paso 2.8

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

$0 = 0$

Paso 3

El dominio de la expresión son todos números reales, excepto cuando la expresión no está definida. En ese caso, no hay ningún número real que haga que la expresión sea indefinida.