Precálculo Ejemplos

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

Paso 1

Comienza por el lado derecho.

${(\sec (x) + \tan (x))}^{2}$

Paso 2

Convierte a senos y cosenos.

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Paso 2.1

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

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

Paso 2.2

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

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

Paso 2.3

Simplifica.

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Paso 2.3.1

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

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

Paso 2.3.2

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 2.3.2.1

Aplica la propiedad distributiva.

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

Paso 2.3.2.2

Aplica la propiedad distributiva.

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

Paso 2.3.2.3

Aplica la propiedad distributiva.

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

Paso 2.3.3

Simplifica y combina los términos similares.

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Paso 2.3.3.1

Simplifica cada término.

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Paso 2.3.3.1.1

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

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Paso 2.3.3.1.1.1

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

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

Paso 2.3.3.1.1.2

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

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

Paso 2.3.3.1.1.3

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

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

Paso 2.3.3.1.1.4

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

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

Paso 2.3.3.1.1.5

Suma $1$ y $1$ .

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

Paso 2.3.3.1.2

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

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Paso 2.3.3.1.2.1

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

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

Paso 2.3.3.1.2.2

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

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

Paso 2.3.3.1.2.3

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

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

Paso 2.3.3.1.2.4

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

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

Paso 2.3.3.1.2.5

Suma $1$ y $1$ .

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

Paso 2.3.3.1.3

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

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Paso 2.3.3.1.3.1

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

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

Paso 2.3.3.1.3.2

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

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

Paso 2.3.3.1.3.3

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

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

Paso 2.3.3.1.3.4

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

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

Paso 2.3.3.1.3.5

Suma $1$ y $1$ .

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

Paso 2.3.3.1.4

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

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Paso 2.3.3.1.4.1

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

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

Paso 2.3.3.1.4.2

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

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

Paso 2.3.3.1.4.3

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

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

Paso 2.3.3.1.4.4

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

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

Paso 2.3.3.1.4.5

Suma $1$ y $1$ .

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

Paso 2.3.3.1.4.6

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

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

Paso 2.3.3.1.4.7

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

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

Paso 2.3.3.1.4.8

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

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

Paso 2.3.3.1.4.9

Suma $1$ y $1$ .

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

Paso 2.3.3.2

Combina los numeradores sobre el denominador común.

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

Paso 2.3.4

Combina los numeradores sobre el denominador común.

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

Paso 2.3.5

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

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

Paso 2.3.6

Factoriza con la regla del cuadrado perfecto.

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

Paso 2.4

Simplifica.

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Paso 2.4.1

Reescribe ${(1 + \sin (x))}^{2}$ como $(1 + \sin (x)) (1 + \sin (x))$ .

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

Paso 2.4.2

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

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Paso 2.4.2.1

Aplica la propiedad distributiva.

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

Paso 2.4.2.2

Aplica la propiedad distributiva.

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

Paso 2.4.2.3

Aplica la propiedad distributiva.

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

Paso 2.4.3

Simplifica y combina los términos similares.

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

Paso 3

Factoriza con la regla del cuadrado perfecto.

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Paso 3.1

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

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

Paso 3.2

Comprueba que el término medio sea dos veces el producto de los números que se elevan al cuadrado en el primer término y el tercer término.

$2 \sin (x) = 2 \cdot 1 \cdot \sin (x)$

Paso 3.3

Reescribe el polinomio.

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

Paso 3.4

Factoriza con la regla del trinomio cuadrado perfecto $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , donde $a = 1$ y $b = \sin (x)$ .

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

Paso 4

Aplica la identidad pitagórica al revés.

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

Paso 5

Simplifica.

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Paso 5.1

Simplifica el denominador.

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Paso 5.1.1

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

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

Paso 5.1.2

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 $b = \sin (x)$ .

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

Paso 5.2

Cancela el factor común de ${(1 + \sin (x))}^{2}$ y $1 + \sin (x)$ .

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

Paso 6

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

$\frac{1 + \sin (x)}{1 - \sin (x)} = {(\sec (x) + \tan (x))}^{2}$ es una identidad