Trigonometría Ejemplos

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

Step 1

Comienza por el lado derecho.

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

Step 2

Convierte a senos y cosenos.

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Escribe $\cot (x)$ en senos y cosenos mediante la identidad del cociente.

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

Aplica la identidad recíproca a $\csc (x)$ .

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

Simplifica.

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

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

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

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Aplica la propiedad distributiva.

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

Aplica la propiedad distributiva.

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

Aplica la propiedad distributiva.

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

Simplifica y combina los términos similares.

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Simplifica cada término.

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

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

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

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

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

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

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

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

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

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

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Multiplica $- 1$ por $- 1$ .

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

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

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

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

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

Combina los numeradores sobre el denominador común.

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

Combina los numeradores sobre el denominador común.

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

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

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

Factoriza con la regla del cuadrado perfecto.

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

Simplifica.

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

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

Expande $(\cos (x) - 1) (\cos (x) - 1)$ con el método PEIU (primero, exterior, interior, ultimo).

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Aplica la propiedad distributiva.

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

Aplica la propiedad distributiva.

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

Aplica la propiedad distributiva.

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

Simplifica y combina los términos similares.

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

Step 3

Factoriza con la regla del cuadrado perfecto.

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

Step 4

Aplica la identidad pitagórica al revés.

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

Step 5

Simplifica.

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Simplifica el denominador.

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Reescribe $1$ como $1^{2}$ .

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

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 $a = 1$ y $b = \cos (x)$ .

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

Cancela el factor común de ${(\cos (x) - 1)}^{2}$ y $1 - \cos (x)$ .

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

Step 6

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

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