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Popular >

verificar (sin^4(x)-1)/(cos^2(x))=cos^2(x)-2

Solución

verificar

\frac{sin ^{4} ( x ) - 1}{cos ^{2} ( x )} = cos^{2} (x) - 2

Solución

Verdadero

Pasos de solución

\frac{sin ^{4} ( x ) - 1}{cos ^{2} ( x )} = cos^{2} (x) - 2

Manipular el lado derecho

\frac{sin ^{4} ( x ) - 1}{cos ^{2} ( x )}

Re-escribir usando identidades trigonométricas

\frac{sin ^{4} ( x ) - 1}{cos ^{2} ( x )}

Utilizar la identidad pitagórica:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= \frac{sin ^{4} ( x ) - 1}{1 - sin ^{2} ( x )}

Simplificar

\frac{sin ^{4} ( x ) - 1}{1 - sin ^{2} ( x )} : - sin^{2} (x) - 1

\frac{sin ^{4} ( x ) - 1}{1 - sin ^{2} ( x )}

Factorizar

sin^{4} (x) - 1 : (sin^{2} (x) + 1) (sin (x) + 1) (sin (x) - 1)

sin^{4} (x) - 1

Reescribir

sin^{4} (x) - 1

como

(sin^{2} (x))^{2} - 1^{2}

sin^{4} (x) - 1

Reescribir

1

como

1^{2}

= sin^{4} (x) - 1^{2}

Aplicar las leyes de los exponentes:

a^{b c} = (a^{b})^{c}

sin^{4} (x) = (sin^{2} (x))^{2}

= (sin^{2} (x))^{2} - 1^{2}

= (sin^{2} (x))^{2} - 1^{2}

Aplicar la siguiente regla para binomios al cuadrado:

x^{2} - y^{2} = (x + y) (x - y)

(sin^{2} (x))^{2} - 1^{2} = (sin^{2} (x) + 1) (sin^{2} (x) - 1)

= (sin^{2} (x) + 1) (sin^{2} (x) - 1)

Factorizar

sin^{2} (x) - 1 : (sin (x) + 1) (sin (x) - 1)

sin^{2} (x) - 1

Reescribir

1

como

1^{2}

= sin^{2} (x) - 1^{2}

Aplicar la siguiente regla para binomios al cuadrado:

x^{2} - y^{2} = (x + y) (x - y)

sin^{2} (x) - 1^{2} = (sin (x) + 1) (sin (x) - 1)

= (sin (x) + 1) (sin (x) - 1)

= (sin^{2} (x) + 1) (sin (x) + 1) (sin (x) - 1)

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

Factorizar

1 - sin^{2} (x) : - (sin (x) + 1) (sin (x) - 1)

1 - sin^{2} (x)

Factorizar el termino común

- 1

= - (sin^{2} (x) - 1)

Factorizar

sin^{2} (x) - 1 : (sin (x) + 1) (sin (x) - 1)

sin^{2} (x) - 1

Reescribir

1

como

1^{2}

= sin^{2} (x) - 1^{2}

Aplicar la siguiente regla para binomios al cuadrado:

x^{2} - y^{2} = (x + y) (x - y)

sin^{2} (x) - 1^{2} = (sin (x) + 1) (sin (x) - 1)

= (sin (x) + 1) (sin (x) - 1)

= - (sin (x) + 1) (sin (x) - 1)

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

Cancelar

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

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

Eliminar los terminos comunes:

sin (x) + 1

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

Eliminar los terminos comunes:

sin (x) - 1

= - (sin^{2} (x) + 1)

= - (sin^{2} (x) + 1)

Negar

- (sin^{2} (x) + 1) = - sin^{2} (x) - 1

= - sin^{2} (x) - 1

= - sin^{2} (x) - 1

= - sin^{2} (x) - 1

Manipular el lado izquierdo

cos^{2} (x) - 2

Re-escribir usando identidades trigonométricas

cos^{2} (x) - 2

Utilizar la identidad pitagórica:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= 1 - sin^{2} (x) - 2

Simplificar

1 - sin^{2} (x) - 2 : - sin^{2} (x) - 1

1 - sin^{2} (x) - 2

Agrupar términos semejantes

= - sin^{2} (x) + 1 - 2

Sumar/restar lo siguiente:

1 - 2 = - 1

= - sin^{2} (x) - 1

= - sin^{2} (x) - 1

= - sin^{2} (x) - 1

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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