Precálculo Ejemplos

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

Paso 1

Comienza por el lado izquierdo.

$\frac{\tan (x) - \cot (x)}{\tan (x) + \cot (x)}$

Paso 2

Simplifica la expresión.

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Paso 2.1

Simplifica el numerador.

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Paso 2.1.1

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

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

Paso 2.1.2

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

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

Paso 2.2

Simplifica el denominador.

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Paso 2.2.1

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

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

Paso 2.2.2

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

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

Paso 2.3

Multiply the numerator and denominator of the fraction by $\cos (x) \sin (x)$ .

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Paso 2.3.1

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

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

Paso 2.3.2

Combinar.

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

Paso 2.4

Aplica la propiedad distributiva.

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

Paso 2.5

Simplifica mediante la cancelación.

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Paso 2.5.1

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

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Paso 2.5.1.1

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

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

Paso 2.5.1.2

Cancela el factor común.

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

Paso 2.5.1.3

Reescribe la expresión.

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

Paso 2.5.2

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

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

Paso 2.5.3

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

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

Paso 2.5.4

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

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

Paso 2.5.5

Suma $1$ y $1$ .

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

Paso 2.5.6

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

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Paso 2.5.6.1

Mueve el signo menos inicial en $- \frac{\cos (x)}{\sin (x)}$ al numerador.

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

Paso 2.5.6.2

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

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

Paso 2.5.6.3

Cancela el factor común.

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

Paso 2.5.6.4

Reescribe la expresión.

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

Paso 2.5.7

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

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

Paso 2.5.8

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

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

Paso 2.5.9

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

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

Paso 2.5.10

Suma $1$ y $1$ .

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

Paso 2.5.11

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

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Paso 2.5.11.1

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

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

Paso 2.5.11.2

Cancela el factor común.

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

Paso 2.5.11.3

Reescribe la expresión.

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

Paso 2.5.12

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

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

Paso 2.5.13

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

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

Paso 2.5.14

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

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

Paso 2.5.15

Suma $1$ y $1$ .

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

Paso 2.5.16

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

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Paso 2.5.16.1

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

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

Paso 2.5.16.2

Cancela el factor común.

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

Paso 2.5.16.3

Reescribe la expresión.

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

Paso 2.5.17

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

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

Paso 2.5.18

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

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

Paso 2.5.19

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

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

Paso 2.5.20

Suma $1$ y $1$ .

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

Paso 2.6

Aplica la identidad pitagórica.

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

Paso 2.7

Divide $\sin^{2} (x) - \cos^{2} (x)$ por $1$ .

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

Paso 3

Aplica la identidad pitagórica al revés.

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

Paso 4

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

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

Paso 5

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

$\frac{\tan (x) - \cot (x)}{\tan (x) + \cot (x)} = 1 - 2 \cos^{2} (x)$ es una identidad