Trigonometría Ejemplos

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

Step 1

Comienza por el lado derecho.

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

Step 2

Convierte a senos y cosenos.

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

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

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

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

Step 3

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

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

Step 4

Combinar.

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

Step 5

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

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

Step 6

Simplifica el denominador.

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

Toca para ver más pasos...

Aplica la propiedad distributiva.

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

Aplica la propiedad distributiva.

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

Aplica la propiedad distributiva.

Simplifica y combina los términos similares.

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Simplifica cada término.

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

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

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

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

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

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

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

Toca para ver más pasos...

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

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

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

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

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

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

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

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

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

Suma $1$ y $1$ .

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

Combina los numeradores sobre el denominador común.

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

Combina los numeradores sobre el denominador común.

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

Suma $\cos (x)$ y $\cos (x)$ .

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

Factoriza con la regla del cuadrado perfecto.

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

Step 7

Aplica la identidad pitagórica al revés.

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

Step 8

Simplifica.

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Multiplica el numerador por la recíproca del denominador.

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

Simplifica el numerador.

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

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

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 $b = \cos (x)$ .

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

Cancela los factores comunes.

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

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

Cancela el factor común.

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

Reescribe la expresión.

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

Aplica la propiedad distributiva.

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

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

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

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

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

Combina los numeradores sobre el denominador común.

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

Simplifica cada término.

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Aplica la propiedad distributiva.

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

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

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

Multiplica $\cos (x) (- \cos (x))$ .

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

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

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

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

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

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

Suma $1$ y $1$ .

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

Suma $- \cos (x)$ y $\cos (x)$ .

$\frac{1 + 0 - {\cos (x)}^{2}}{(1 + \cos (x)) \sin (x)}$

Suma $1$ y $0$ .

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

Simplifica el numerador.

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

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

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 $b = \cos (x)$ .

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

Cancela el factor común de $1 + \cos (x)$ .

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

Step 9

Ahora considera el lado izquierdo de la ecuación.

$\csc (x) - \cot (x)$

Step 10

Convierte a senos y cosenos.

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

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

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

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

Step 11

Combina los numeradores sobre el denominador común.

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

Step 12

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

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