Precálculo Ejemplos

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

Paso 1

Comienza por el lado izquierdo.

$\frac{1}{\sec (x) + \tan (x)}$

Paso 2

Convierte a senos y cosenos.

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Paso 2.1

Aplica la identidad recíproca a $\sec (x)$ .

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

Paso 2.2

Escribe $\tan (x)$ en senos y cosenos mediante la identidad del cociente.

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

Paso 3

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

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

Paso 4

Combinar.

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

Paso 5

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

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

Paso 6

Simplifica el denominador.

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Paso 6.1

Expande $(\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)}) (\frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)})$ con el método PEIU (primero, exterior, interior, ultimo).

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Paso 6.1.1

Aplica la propiedad distributiva.

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

Paso 6.1.2

Aplica la propiedad distributiva.

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

Paso 6.1.3

Aplica la propiedad distributiva.

Paso 6.2

Simplifica y combina los términos similares.

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Paso 6.2.1

Simplifica cada término.

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Paso 6.2.1.1

Multiplica $\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)}$ .

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Paso 6.2.1.1.1

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

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

Paso 6.2.1.1.2

Eleva $\cos (x)$ a la potencia de $1$ .

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

Paso 6.2.1.1.3

Eleva $\cos (x)$ a la potencia de $1$ .

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

Paso 6.2.1.1.4

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Paso 6.2.1.1.5

Suma $1$ y $1$ .

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

Paso 6.2.1.2

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

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Paso 6.2.1.2.1

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

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

Paso 6.2.1.2.2

Eleva $\cos (x)$ a la potencia de $1$ .

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

Paso 6.2.1.2.3

Eleva $\cos (x)$ a la potencia de $1$ .

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

Paso 6.2.1.2.4

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Paso 6.2.1.2.5

Suma $1$ y $1$ .

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

Paso 6.2.1.3

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

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Paso 6.2.1.3.1

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

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

Paso 6.2.1.3.2

Eleva $\cos (x)$ a la potencia de $1$ .

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

Paso 6.2.1.3.3

Eleva $\cos (x)$ a la potencia de $1$ .

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

Paso 6.2.1.3.4

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Paso 6.2.1.3.5

Suma $1$ y $1$ .

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

Paso 6.2.1.4

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

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Paso 6.2.1.4.1

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

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

Paso 6.2.1.4.2

Eleva $\sin (x)$ a la potencia de $1$ .

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

Paso 6.2.1.4.3

Eleva $\sin (x)$ a la potencia de $1$ .

Paso 6.2.1.4.4

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Paso 6.2.1.4.5

Suma $1$ y $1$ .

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

Paso 6.2.1.4.6

Eleva $\cos (x)$ a la potencia de $1$ .

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

Paso 6.2.1.4.7

Eleva $\cos (x)$ a la potencia de $1$ .

Paso 6.2.1.4.8

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Paso 6.2.1.4.9

Suma $1$ y $1$ .

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

Paso 6.2.2

Combina los numeradores sobre el denominador común.

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

Paso 6.3

Combina los numeradores sobre el denominador común.

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

Paso 6.4

Suma $\sin (x)$ y $\sin (x)$ .

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

Paso 6.5

Factoriza con la regla del cuadrado perfecto.

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

Paso 7

Aplica la identidad pitagórica al revés.

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

Paso 8

Simplifica.

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Paso 8.1

Multiplica el numerador por la recíproca del denominador.

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

Paso 8.2

Simplifica el numerador.

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Paso 8.2.1

Reescribe $1$ como $1^{2}$ .

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

Paso 8.2.2

Dado que ambos términos son cuadrados perfectos, factoriza con la fórmula de la diferencia de cuadrados, $a^{2} - b^{2} = (a + b) (a - b)$ , donde $a = 1$ y $b = \sin (x)$ .

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

Paso 8.3

Cancela los factores comunes.

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Paso 8.3.1

Factoriza $1 + \sin (x)$ de ${(1 + \sin (x))}^{2}$ .

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

Paso 8.3.2

Cancela el factor común.

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

Paso 8.3.3

Reescribe la expresión.

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

Paso 8.4

Aplica la propiedad distributiva.

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

Paso 8.5

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

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

Paso 8.6

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

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

Paso 8.7

Combina los numeradores sobre el denominador común.

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

Paso 8.8

Simplifica cada término.

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Paso 8.8.1

Aplica la propiedad distributiva.

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

Paso 8.8.2

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

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

Paso 8.8.3

Multiplica $\sin (x) (- \sin (x))$ .

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Paso 8.8.3.1

Eleva $\sin (x)$ a la potencia de $1$ .

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

Paso 8.8.3.2

Eleva $\sin (x)$ a la potencia de $1$ .

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

Paso 8.8.3.3

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Paso 8.8.3.4

Suma $1$ y $1$ .

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

Paso 8.9

Suma $- \sin (x)$ y $\sin (x)$ .

$\frac{1 + 0 - {\sin (x)}^{2}}{(1 + \sin (x)) \cos (x)}$

Paso 8.10

Suma $1$ y $0$ .

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

Paso 8.11

Simplifica el numerador.

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Paso 8.11.1

Reescribe $1$ como $1^{2}$ .

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

Paso 8.11.2

Dado que ambos términos son cuadrados perfectos, factoriza con la fórmula de la diferencia de cuadrados, $a^{2} - b^{2} = (a + b) (a - b)$ , donde $a = 1$ y $b = \sin (x)$ .

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

Paso 8.12

Cancela el factor común de $1 + \sin (x)$ .

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

Paso 9

Debido a que se ha demostrado que los dos lados son equivalentes, la ecuación es una identidad.

$\frac{1}{\sec (x) + \tan (x)} = \frac{1 - \sin (x)}{\cos (x)}$ es una identidad