Precálculo Ejemplos

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

Paso 1

Simplifica el numerador.

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Paso 1.1

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

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

Paso 1.2

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

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

Paso 1.3

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

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Paso 1.3.1

Aplica la propiedad distributiva.

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

Paso 1.3.2

Aplica la propiedad distributiva.

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

Paso 1.3.3

Aplica la propiedad distributiva.

$\frac{\sin (x) \sin (x) + \sin (x) \frac{\sin (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} \sin (x) + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + \cos^{2} (x) - \sec^{2} (x)}{\tan (x)}$

Paso 1.4

Simplifica y combina los términos similares.

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Paso 1.4.1

Simplifica cada término.

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Paso 1.4.1.1

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

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Paso 1.4.1.1.1

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

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

Paso 1.4.1.1.2

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

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

Paso 1.4.1.1.3

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

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

Paso 1.4.1.1.4

Suma $1$ y $1$ .

$\frac{\sin^{2} (x) + \sin (x) \frac{\sin (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} \sin (x) + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + \cos^{2} (x) - \sec^{2} (x)}{\tan (x)}$

Paso 1.4.1.2

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

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Paso 1.4.1.2.1

Combina $\sin (x)$ y $\frac{\sin (x)}{\cos (x)}$ .

$\frac{\sin^{2} (x) + \frac{\sin (x) \sin (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} \sin (x) + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + \cos^{2} (x) - \sec^{2} (x)}{\tan (x)}$

Paso 1.4.1.2.2

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

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

Paso 1.4.1.2.3

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

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

Paso 1.4.1.2.4

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

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

Paso 1.4.1.2.5

Suma $1$ y $1$ .

$\frac{\sin^{2} (x) + \frac{\sin^{2} (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} \sin (x) + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + \cos^{2} (x) - \sec^{2} (x)}{\tan (x)}$

Paso 1.4.1.3

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

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Paso 1.4.1.3.1

Combina $\frac{\sin (x)}{\cos (x)}$ y $\sin (x)$ .

$\frac{\sin^{2} (x) + \frac{\sin^{2} (x)}{\cos (x)} + \frac{\sin (x) \sin (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + \cos^{2} (x) - \sec^{2} (x)}{\tan (x)}$

Paso 1.4.1.3.2

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

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

Paso 1.4.1.3.3

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

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

Paso 1.4.1.3.4

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

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

Paso 1.4.1.3.5

Suma $1$ y $1$ .

$\frac{\sin^{2} (x) + \frac{\sin^{2} (x)}{\cos (x)} + \frac{\sin^{2} (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + \cos^{2} (x) - \sec^{2} (x)}{\tan (x)}$

Paso 1.4.1.4

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

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Paso 1.4.1.4.1

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

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

Paso 1.4.1.4.2

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

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

Paso 1.4.1.4.3

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

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

Paso 1.4.1.4.4

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

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

Paso 1.4.1.4.5

Suma $1$ y $1$ .

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

Paso 1.4.1.4.6

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

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

Paso 1.4.1.4.7

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

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

Paso 1.4.1.4.8

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

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

Paso 1.4.1.4.9

Suma $1$ y $1$ .

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

Paso 1.4.2

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

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

Paso 1.5

Combina $2$ y $\frac{\sin^{2} (x)}{\cos (x)}$ .

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

Paso 1.6

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

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

Paso 1.7

Aplica la regla del producto a $\frac{1}{\cos (x)}$ .

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

Paso 1.8

Uno elevado a cualquier potencia es uno.

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

Paso 1.9

Reescribe $\sin^{2} (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{2} (x)}{\cos^{2} (x)} + \cos^{2} (x) - \frac{1}{\cos^{2} (x)}$ en forma factorizada.

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Paso 1.9.1

Sea $u = \frac{2 \sin^{2} (x)}{\cos (x)}$ . Sustituye $u$ por todos los casos de $\frac{2 \sin^{2} (x)}{\cos (x)}$ .

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Paso 1.9.1.1

Resta $1$ de $1$ .

$\frac{u + 0}{\tan (x)}$

Paso 1.9.1.2

Suma $u$ y $0$ .

$\frac{u}{\tan (x)}$

Paso 1.9.2

Reemplaza todos los casos de $u$ con $\frac{2 \sin^{2} (x)}{\cos (x)}$ .

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

Paso 2

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

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

Paso 3

Multiplica el numerador por la recíproca del denominador.

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

Paso 4

Combinar.

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

Paso 5

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

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Paso 5.1

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

$\frac{\sin (x) (2 \sin (x) \cos (x))}{\cos (x) \sin (x)}$

Paso 5.2

Cancela los factores comunes.

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Paso 5.2.1

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

$\frac{\sin (x) (2 \sin (x) \cos (x))}{\sin (x) \cos (x)}$

Paso 5.2.2

Cancela el factor común.

$\frac{\sin (x) (2 \sin (x) \cos (x))}{\sin (x) \cos (x)}$

Paso 5.2.3

Reescribe la expresión.

$\frac{2 \sin (x) \cos (x)}{\cos (x)}$

Paso 6

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

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Paso 6.1

Cancela el factor común.

$\frac{2 \sin (x) \cos (x)}{\cos (x)}$

Paso 6.2

Divide $2 \sin (x)$ por $1$ .

$2 \sin (x)$