Trigonometría Ejemplos

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

Step 1

Comienza por el lado izquierdo.

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

Step 2

Convierte a senos y cosenos.

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

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

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

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

Step 3

Simplifica.

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Multiplica el numerador y el denominador de la fracción compleja por $\cos (x) \sin (x)$ .

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

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

Combinar.

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

Aplica la propiedad distributiva.

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

Simplifica mediante la cancelación.

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

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

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

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

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

Cancela el factor común.

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

Reescribe la expresión.

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

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

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

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

Cancela el factor común.

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

Reescribe la expresión.

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

Simplifica el numerador.

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Factoriza ${\sin (x)}^{3}$ de $\cos (x) \sin (x) {\sin (x)}^{2} - {\sin (x)}^{2}$ .

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Factoriza ${\sin (x)}^{3}$ de $\cos (x) \sin (x) {\sin (x)}^{2}$ .

$\frac{{\sin (x)}^{3} (\cos (x) \sin (x) {\sin (x)}^{- 1}) - {\sin (x)}^{2}}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Factoriza ${\sin (x)}^{3}$ de $- {\sin (x)}^{2}$ .

$\frac{{\sin (x)}^{3} (\cos (x) \sin (x) {\sin (x)}^{- 1}) + {\sin (x)}^{3} (- {\sin (x)}^{- 1})}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Factoriza ${\sin (x)}^{3}$ de ${\sin (x)}^{3} (\cos (x) \sin (x) {\sin (x)}^{- 1}) + {\sin (x)}^{3} (- {\sin (x)}^{- 1})$ .

$\frac{{\sin (x)}^{3} (\cos (x) \sin (x) {\sin (x)}^{- 1} - {\sin (x)}^{- 1})}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

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

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

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\cos (x) \sin (x) \sin (x)) - {\sin (x)}^{- 1})}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Factoriza ${\sin (x)}^{- 2}$ de $- {\sin (x)}^{- 1}$ .

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\cos (x) \sin (x) \sin (x)) + {\sin (x)}^{- 2} (- \sin (x)))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

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

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\cos (x) \sin (x) \sin (x) - \sin (x)))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

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

Toca para ver más pasos...

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

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\cos (x) ({\sin (x)}^{1} \sin (x)) - \sin (x)))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

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

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\cos (x) ({\sin (x)}^{1} {\sin (x)}^{1}) - \sin (x)))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

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

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\cos (x) {\sin (x)}^{1 + 1} - \sin (x)))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Suma $1$ y $1$ .

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\cos (x) {\sin (x)}^{2} - \sin (x)))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

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

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

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\sin (x) (\cos (x) \sin (x)) - \sin (x)))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

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

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\sin (x) (\cos (x) \sin (x)) + \sin (x) \cdot - 1))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Factoriza $\sin (x)$ de $\sin (x) (\cos (x) \sin (x)) + \sin (x) \cdot - 1$ .

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} (\sin (x) (\cos (x) \sin (x) - 1)))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} \sin (x) (\cos (x) \sin (x) - 1))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Multiplica ${\sin (x)}^{- 2}$ por $\sin (x)$ sumando los exponentes.

Toca para ver más pasos...

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

Toca para ver más pasos...

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

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2} {\sin (x)}^{1} (\cos (x) \sin (x) - 1))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

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

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 2 + 1} (\cos (x) \sin (x) - 1))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Suma $- 2$ y $1$ .

$\frac{{\sin (x)}^{3} ({\sin (x)}^{- 1} (\cos (x) \sin (x) - 1))}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

$\frac{{\sin (x)}^{3} {\sin (x)}^{- 1} (\cos (x) \sin (x) - 1)}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Multiplica ${\sin (x)}^{3}$ por ${\sin (x)}^{- 1}$ sumando los exponentes.

Toca para ver más pasos...

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

$\frac{{\sin (x)}^{3 - 1} (\cos (x) \sin (x) - 1)}{\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}}$

Resta $1$ de $3$ .

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

Simplifica el denominador.

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Factoriza ${\cos (x)}^{3}$ de $\cos (x) \sin (x) {\cos (x)}^{2} - {\cos (x)}^{2}$ .

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Factoriza ${\cos (x)}^{3}$ de $\cos (x) \sin (x) {\cos (x)}^{2}$ .

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} (\cos (x) \sin (x) {\cos (x)}^{- 1}) - {\cos (x)}^{2}}$

Factoriza ${\cos (x)}^{3}$ de $- {\cos (x)}^{2}$ .

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} (\cos (x) \sin (x) {\cos (x)}^{- 1}) + {\cos (x)}^{3} (- {\cos (x)}^{- 1})}$

Factoriza ${\cos (x)}^{3}$ de ${\cos (x)}^{3} (\cos (x) \sin (x) {\cos (x)}^{- 1}) + {\cos (x)}^{3} (- {\cos (x)}^{- 1})$ .

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} (\cos (x) \sin (x) {\cos (x)}^{- 1} - {\cos (x)}^{- 1})}$

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

Toca para ver más pasos...

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} (\cos (x) \sin (x) \cos (x)) - {\cos (x)}^{- 1})}$

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} (\cos (x) \sin (x) \cos (x)) + {\cos (x)}^{- 2} (- \cos (x)))}$

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} (\cos (x) \sin (x) \cos (x) - \cos (x)))}$

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

Toca para ver más pasos...

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} ({\cos (x)}^{1} \cos (x) \sin (x) - \cos (x)))}$

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} ({\cos (x)}^{1} {\cos (x)}^{1} \sin (x) - \cos (x)))}$

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} ({\cos (x)}^{1 + 1} \sin (x) - \cos (x)))}$

Suma $1$ y $1$ .

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} ({\cos (x)}^{2} \sin (x) - \cos (x)))}$

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

Toca para ver más pasos...

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} (\cos (x) (\cos (x) \sin (x)) - \cos (x)))}$

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} (\cos (x) (\cos (x) \sin (x)) + \cos (x) \cdot - 1))}$

Factoriza $\cos (x)$ de $\cos (x) (\cos (x) \sin (x)) + \cos (x) \cdot - 1$ .

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} (\cos (x) (\cos (x) \sin (x) - 1)))}$

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} \cos (x) (\cos (x) \sin (x) - 1))}$

Multiplica ${\cos (x)}^{- 2}$ por $\cos (x)$ sumando los exponentes.

Toca para ver más pasos...

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

Toca para ver más pasos...

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2} {\cos (x)}^{1} (\cos (x) \sin (x) - 1))}$

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 2 + 1} (\cos (x) \sin (x) - 1))}$

Suma $- 2$ y $1$ .

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} ({\cos (x)}^{- 1} (\cos (x) \sin (x) - 1))}$

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3} {\cos (x)}^{- 1} (\cos (x) \sin (x) - 1)}$

Multiplica ${\cos (x)}^{3}$ por ${\cos (x)}^{- 1}$ sumando los exponentes.

Toca para ver más pasos...

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

$\frac{{\sin (x)}^{2} (\cos (x) \sin (x) - 1)}{{\cos (x)}^{3 - 1} (\cos (x) \sin (x) - 1)}$

Resta $1$ de $3$ .

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

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

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

Step 4

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

$\tan^{2} (x)$

Step 5

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

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