Trigonometría Ejemplos

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

Step 1

Comienza por el lado izquierdo.

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

Step 2

Convierte a senos y cosenos.

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

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

Aplica la regla del producto a $\frac{\sin (x)}{\cos (x)}$ .

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

Step 3

Simplifica.

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Multiplica el numerador y el denominador de la fracción compleja por ${\cos (x)}^{2}$ .

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

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

Combinar.

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

Aplica la propiedad distributiva.

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

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

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Mueve el signo menos inicial en $- \frac{{\sin (x)}^{2}}{{\cos (x)}^{2}}$ al numerador.

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

Cancela el factor común.

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

Reescribe la expresión.

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

Simplifica el numerador.

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

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

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

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

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

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

Simplifica.

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

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

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

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

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

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

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

Suma $1$ y $1$ .

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

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

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

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

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

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

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

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

Combina exponentes.

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

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

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

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

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

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

Suma $1$ y $1$ .

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

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

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

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

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

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

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

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

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

Combina exponentes.

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

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

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

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

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

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

Suma $1$ y $1$ .

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

Simplifica el denominador.

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

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

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

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

Simplifica.

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

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

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

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

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

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Cancela el factor común.

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

Reescribe la expresión.

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

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

$\cos^{2} (x)$

Step 4

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

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