Álgebra Ejemplos

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

Paso 1

Simplifica con la obtención del factor común.

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Paso 1.1

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

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

Paso 1.2

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

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

Paso 1.3

Factoriza $\cot (x)$ de $\cot (x) (\sec^{2} (x)) + \cot (x) \cdot - 1$ .

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

Paso 2

Aplica la identidad pitagórica.

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

Paso 3

Simplifica el numerador.

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Paso 3.1

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

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

Paso 3.2

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

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

Paso 3.3

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

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

Paso 4

Simplifica el denominador.

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Paso 4.1

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

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

Paso 4.2

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

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Paso 4.2.1

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

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

Paso 4.2.2

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

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

Paso 4.2.3

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

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

Paso 4.2.4

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

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

Paso 4.2.5

Suma $1$ y $1$ .

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

Paso 5

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

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

Paso 6

Simplifica el numerador.

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Paso 6.1

Reduce la expresión $\frac{\cos (x) \sin^{2} (x)}{\sin (x) \cos^{2} (x)}$ mediante la cancelación de los factores comunes.

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Paso 6.1.1

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

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

Paso 6.1.2

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

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

Paso 6.1.3

Cancela el factor común.

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

Paso 6.1.4

Reescribe la expresión.

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

Paso 6.2

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

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Paso 6.2.1

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

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

Paso 6.2.2

Cancela los factores comunes.

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Paso 6.2.2.1

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

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

Paso 6.2.2.2

Cancela el factor común.

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

Paso 6.2.2.3

Reescribe la expresión.

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

Paso 7

Multiplica el numerador por la recíproca del denominador.

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

Paso 8

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

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

Paso 9

Separa las fracciones.

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

Paso 10

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

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

Paso 11

Simplifica cada término.

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Paso 11.1

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

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

Paso 11.2

Separa las fracciones.

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

Paso 11.3

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

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

Paso 11.4

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

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

Paso 12

Combina $\frac{1}{\sin (x) \tan (x) + \cos (x)}$ y $\tan (x)$ .

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