Trigonometría Ejemplos

\frac{{csc}^{2} (x)}{{cot}^{2} (x) - 1} = \frac{{sec}^{2} (x)}{1 - {tan}^{2} (x)}

$\frac{\csc^{2} (x)}{\cot^{2} (x) - 1} = \frac{\sec^{2} (x)}{1 - \tan^{2} (x)}$

Step 1

Comienza por el lado izquierdo.

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

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

Step 2

Convierte a senos y cosenos.

Toca para ver más pasos...

Aplica la identidad recíproca a

csc (x)

$\csc (x)$ .

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

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

Escribe

cot (x)

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

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

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

Aplica la regla del producto a

\frac{1}{sin (x)}

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

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

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

Aplica la regla del producto a

\frac{cos (x)}{sin (x)}

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

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

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

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

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

Step 3

Simplifica.

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Multiplica el numerador por la recíproca del denominador.

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

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

Uno elevado a cualquier potencia es uno.

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

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

Simplifica el denominador.

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Reescribe

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

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

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

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

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

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

Reescribe

1

$1$ como

1^{2}

$1^{2}$ .

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

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

Dado que ambos términos son cuadrados perfectos, factoriza con la fórmula de la diferencia de cuadrados,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ , donde

a = \frac{cos (x)}{sin (x)}

$a = \frac{\cos (x)}{\sin (x)}$ y

b = 1

$b = 1$ .

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

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

Escribe

1

$1$ como una fracción con un denominador común.

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

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

Combina los numeradores sobre el denominador común.

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

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

Para escribir

- 1

$- 1$ como una fracción con un denominador común, multiplica por

\frac{sin (x)}{sin (x)}

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

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

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

Combina

- 1

$- 1$ y

\frac{sin (x)}{sin (x)}

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

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

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

Combina los numeradores sobre el denominador común.

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

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

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

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

Multiplica

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

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

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

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

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

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

Simplifica el denominador.

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Eleva

sin (x)

$\sin (x)$ a la potencia de

1

$1$ .

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

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

Eleva

sin (x)

$\sin (x)$ a la potencia de

1

$1$ .

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

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

Usa la regla de la potencia

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

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

Suma

1

$1$ y

1

$1$ .

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

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

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

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

Combinar.

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

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

Multiplica

1

$1$ por

1

$1$ .

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

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

Combina

{sin (x)}^{2}

${\sin (x)}^{2}$ y

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

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

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

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

Reduce la expresión mediante la cancelación de los factores comunes.

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

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

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

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

Step 4

Reescribe

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

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

$\frac{\sec^{2} (x)}{1 - \tan^{2} (x)}$ .

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

Step 5

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

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