Precálculo Ejemplos

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

Paso 1

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

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

Paso 2

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

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

Paso 3

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

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Paso 3.1

Aplica la propiedad distributiva.

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

Paso 3.2

Aplica la propiedad distributiva.

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

Paso 3.3

Aplica la propiedad distributiva.

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

Paso 4

Simplifica y combina los términos similares.

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Paso 4.1

Simplifica cada término.

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Paso 4.1.1

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

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Paso 4.1.1.1

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

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

Paso 4.1.1.2

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

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

Paso 4.1.1.3

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

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

Paso 4.1.1.4

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

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

Paso 4.1.1.5

Suma $1$ y $1$ .

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

Paso 4.1.1.6

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

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

Paso 4.1.1.7

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

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

Paso 4.1.1.8

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

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

Paso 4.1.1.9

Suma $1$ y $1$ .

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

Paso 4.1.2

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

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

Paso 4.1.3

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

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

Paso 4.1.4

Multiplica $1$ por $1$ .

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

Paso 4.2

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

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

Paso 5

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

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

Paso 6

Aplica la propiedad distributiva.

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

Paso 7

Simplifica.

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Paso 7.1

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

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Paso 7.1.1

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

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

Paso 7.1.2

Cancela el factor común.

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

Paso 7.1.3

Reescribe la expresión.

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

Paso 7.2

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

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Paso 7.2.1

Cancela el factor común.

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

Paso 7.2.2

Reescribe la expresión.

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

Paso 7.3

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

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

Paso 8

Simplifica cada término.

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Paso 8.1

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

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

Paso 8.2

Separa las fracciones.

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

Paso 8.3

Convierte de $\frac{\sin (x)}{\cos (x)}$ a $\tan (x)$ .

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

Paso 8.4

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

$\sin (x) \tan (x) + 2 \sin (x) + \cos (x)$