Precálculo Ejemplos

$\sin (x + y) \cos (x - y) = \frac{\sin (x)}{\sec (x)} + \frac{\cos (y)}{\csc (y)}$

Paso 1

Comienza por el lado izquierdo.

$\sin (x + y) \cos (x - y)$

Paso 2

Aplica la suma de la razón de los ángulos.

$(\sin (x) \cos (y) + \cos (x) \sin (y)) \cos (x - y)$

Paso 3

Aplica la suma de la razón de los ángulos $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

$(\sin (x) \cos (y) + \cos (x) \sin (y)) (\cos (x) \cos (- y) - \sin (x) \sin (- y))$

Paso 4

Simplifica la expresión.

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Paso 4.1

Simplifica cada término.

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Paso 4.1.1

Como $\cos (- y)$ es una función par, reescribe $\cos (- y)$ como $\cos (y)$ .

$(\sin (x) \cos (y) + \cos (x) \sin (y)) (\cos (x) \cos (y) - \sin (x) \sin (- y))$

Paso 4.1.2

Como $\sin (- y)$ es una función impar, reescribe $\sin (- y)$ como $- \sin (y)$ .

$(\sin (x) \cos (y) + \cos (x) \sin (y)) (\cos (x) \cos (y) - \sin (x) (- \sin (y)))$

Paso 4.1.3

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

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Paso 4.1.3.1

Multiplica $- 1$ por $- 1$ .

$(\sin (x) \cos (y) + \cos (x) \sin (y)) (\cos (x) \cos (y) + 1 \sin (x) \sin (y))$

Paso 4.1.3.2

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

$(\sin (x) \cos (y) + \cos (x) \sin (y)) (\cos (x) \cos (y) + \sin (x) \sin (y))$

Paso 4.2

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

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Paso 4.2.1

Aplica la propiedad distributiva.

$\sin (x) \cos (y) (\cos (x) \cos (y) + \sin (x) \sin (y)) + \cos (x) \sin (y) (\cos (x) \cos (y) + \sin (x) \sin (y))$

Paso 4.2.2

Aplica la propiedad distributiva.

$\sin (x) \cos (y) (\cos (x) \cos (y)) + \sin (x) \cos (y) (\sin (x) \sin (y)) + \cos (x) \sin (y) (\cos (x) \cos (y) + \sin (x) \sin (y))$

Paso 4.2.3

Aplica la propiedad distributiva.

$\sin (x) \cos (y) (\cos (x) \cos (y)) + \sin (x) \cos (y) (\sin (x) \sin (y)) + \cos (x) \sin (y) (\cos (x) \cos (y)) + \cos (x) \sin (y) (\sin (x) \sin (y))$

Paso 4.3

Simplifica cada término.

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Paso 4.3.1

Multiplica $\sin (x) \cos (y) (\cos (x) \cos (y))$ .

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Paso 4.3.1.1

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

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

Paso 4.3.1.2

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

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

Paso 4.3.1.3

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

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

Paso 4.3.1.4

Suma $1$ y $1$ .

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

Paso 4.3.2

Multiplica $\sin (x) \cos (y) (\sin (x) \sin (y))$ .

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Paso 4.3.2.1

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

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

Paso 4.3.2.2

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

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

Paso 4.3.2.3

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

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

Paso 4.3.2.4

Suma $1$ y $1$ .

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

Paso 4.3.3

Multiplica $\cos (x) \sin (y) (\cos (x) \cos (y))$ .

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Paso 4.3.3.1

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

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

Paso 4.3.3.2

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

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

Paso 4.3.3.3

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

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

Paso 4.3.3.4

Suma $1$ y $1$ .

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

Paso 4.3.4

Multiplica $\cos (x) \sin (y) (\sin (x) \sin (y))$ .

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Paso 4.3.4.1

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

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

Paso 4.3.4.2

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

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

Paso 4.3.4.3

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

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

Paso 4.3.4.4

Suma $1$ y $1$ .

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

Paso 5

Factoriza.

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Paso 5.1

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

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Paso 5.1.1

Factoriza $\sin (x) \cos (y)$ de $\sin (x) \cos^{2} (y) \cos (x)$ .

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

Paso 5.1.2

Factoriza $\sin (x) \cos (y)$ de $\sin^{2} (x) \cos (y) \sin (y)$ .

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

Paso 5.1.3

Factoriza $\sin (x) \cos (y)$ de $\sin (x) \cos (y) (\cos (y) \cos (x)) + \sin (x) \cos (y) (\sin (x) \sin (y))$ .

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

Paso 5.2

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

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Paso 5.2.1

Factoriza $\cos (y)$ de $\sin (x) \cos (y) (\cos (y) \cos (x) + \sin (x) \sin (y))$ .

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

Paso 5.2.2

Factoriza $\cos (y)$ de $\cos^{2} (x) \sin (y) \cos (y)$ .

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

Paso 5.2.3

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

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

Paso 5.3

Aplica la propiedad distributiva.

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

Paso 5.4

Elimina los paréntesis.

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

Paso 5.5

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

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Paso 5.5.1

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

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

Paso 5.5.2

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

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

Paso 5.5.3

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

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

Paso 5.5.4

Suma $1$ y $1$ .

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

Paso 5.6

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

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Paso 5.6.1

Factoriza $\sin (x)$ de $\sin (x) \cos (y) \cos (x)$ .

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

Paso 5.6.2

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

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

Paso 5.6.3

Factoriza $\sin (x)$ de $\sin (x) (\cos (y) \cos (x)) + \sin (x) (\sin (x) \sin (y))$ .

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

Paso 6

Aplica la identidad pitagórica al revés.

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

Paso 7

Aplica la propiedad distributiva.

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

Paso 8

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

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

Paso 9

Aplica la propiedad distributiva.

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

Paso 10

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

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

Paso 11

Aplica la propiedad distributiva.

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

Paso 12

Simplifica.

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Paso 12.1

Multiplica $\cos (y) (\sin (x) \cos (y) \cos (x))$ .

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Paso 12.1.1

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

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

Paso 12.1.2

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

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

Paso 12.1.3

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

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

Paso 12.1.4

Suma $1$ y $1$ .

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

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

Paso 12.2

Suma $\cos (y) {\sin (x)}^{2} \sin (y)$ y $\cos (y) (- {\sin (x)}^{2} \sin (y))$ .

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Paso 12.2.1

Reordena $\cos (y)$ y $- 1$ .

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

Paso 12.2.2

Resta $\cos (y) {\sin (x)}^{2} \sin (y)$ de $\cos (y) {\sin (x)}^{2} \sin (y)$ .

${\cos (y)}^{2} \sin (x) \cos (x) + \cos (y) \sin (y) + 0 + \cos (x) \sin^{2} (y) \sin (x)$

Paso 12.3

Suma ${\cos (y)}^{2} \sin (x) \cos (x)$ y $0$ .

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

Paso 13

Aplica la identidad pitagórica al revés.

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

Paso 14

Simplifica cada término.

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

Paso 15

Simplifica.

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Paso 15.1

Suma $- {\sin (y)}^{2} \cos (x) \sin (x)$ y ${\sin (y)}^{2} \cos (x) \sin (x)$ .

$\cos (x) \sin (x) + \cos (y) \sin (y) + 0$

Paso 15.2

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

$\cos (x) \sin (x) + \cos (y) \sin (y)$

Paso 16

Reescribe $\cos (x) \sin (x) + \cos (y) \sin (y)$ como $\frac{\sin (x)}{\sec (x)} + \frac{\cos (y)}{\csc (y)}$ .

$\frac{\sin (x)}{\sec (x)} + \frac{\cos (y)}{\csc (y)}$

Paso 17

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

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