Álgebra Ejemplos

$\frac{\cot^{2} (x) \cos^{2} (x)}{\cot^{2} (x) - \cos^{2} (x)}$

Paso 1

Simplifica el numerador.

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Paso 1.1

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

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

Paso 1.2

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

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

Paso 2

Simplifica el denominador.

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Paso 2.1

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 $a = \cot (x)$ y $b = \cos (x)$ .

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

Paso 2.2

Simplifica.

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Paso 2.2.1

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

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

Paso 2.2.2

Factoriza $\cos (x)$ de $\frac{\cos (x)}{\sin (x)} + \cos (x)$ .

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Paso 2.2.2.1

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

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

Paso 2.2.2.2

Multiplica por $1$ .

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

Paso 2.2.2.3

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

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

Paso 2.2.3

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

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

Paso 2.2.4

Factoriza $\cos (x)$ de $\frac{\cos (x)}{\sin (x)} - \cos (x)$ .

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Paso 2.2.4.1

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

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

Paso 2.2.4.2

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

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

Paso 2.2.4.3

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

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

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

Paso 2.2.5

Combina exponentes.

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Paso 2.2.5.1

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

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

Paso 2.2.5.2

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

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

Paso 2.2.5.3

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

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

Paso 2.2.5.4

Suma $1$ y $1$ .

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

Paso 3

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

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

Paso 4

Multiplica $\cos^{2} (x)$ por $\cos^{2} (x)$ sumando los exponentes.

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Paso 4.1

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

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

Paso 4.2

Suma $2$ y $2$ .

$\frac{\frac{\cos^{4} (x)}{\sin^{2} (x)}}{\cos^{2} (x) (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1)}$

Paso 5

Multiplica el numerador por la recíproca del denominador.

$\frac{\cos^{4} (x)}{\sin^{2} (x)} \cdot \frac{1}{\cos^{2} (x) (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1)}$

Paso 6

Combinar.

$\frac{\cos^{4} (x) \cdot 1}{\sin^{2} (x) (\cos^{2} (x) (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1))}$

Paso 7

Cancela el factor común de $\cos^{4} (x)$ y $\cos^{2} (x)$ .

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Paso 7.1

Factoriza $\cos^{2} (x)$ de $\cos^{4} (x) \cdot 1$ .

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

Paso 7.2

Cancela los factores comunes.

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Paso 7.2.1

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

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

Paso 7.2.2

Cancela el factor común.

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

Paso 7.2.3

Reescribe la expresión.

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

Paso 8

Multiplica $\cos^{2} (x)$ por $1$ .

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

Paso 9

Multiplica por $1$ .

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

Paso 10

Multiplica por $1$ .

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

Paso 11

Separa las fracciones.

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

Paso 12

Convierte de $\frac{\cos^{2} (x)}{\sin^{2} (x)}$ a $\cot^{2} (x)$ .

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

Paso 13

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

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

Paso 14

Simplifica el denominador.

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Paso 14.1

Convierte de $\frac{1}{\sin (x)}$ a $\csc (x)$ .

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

Paso 14.2

Convierte de $\frac{1}{\sin (x)}$ a $\csc (x)$ .

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

Paso 15

Combina $\frac{1}{(\csc (x) + 1) (\csc (x) - 1)}$ y $\cot^{2} (x)$ .

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