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verificar 1/((1-\frac{1){(1-1/((1-cosh^2(x))))})}=cosh^2(x)

Solución

verificar

\frac{1}{1 - \frac{1}{( 1 - \frac{1}{( 1 - c o s h ^{2} ( x ) )} )}} = cosh^{2} (x)

Verdadero

Pasos de solución

\frac{1}{1 - \frac{1}{1 - \frac{1}{1 - c o s h ^{2} ( x )}}} = cosh^{2} (x)

Manipular el lado derecho

\frac{1}{1 - \frac{1}{1 - \frac{1}{1 - c o s h ^{2} ( x )}}}

Simplificar

\frac{1}{1 - \frac{1}{1 - \frac{1}{1 - c o s h ^{2} ( x )}}} : cosh^{2} (x)

\frac{1}{1 - \frac{1}{1 - \frac{1}{1 - c o s h ^{2} ( x )}}}

\frac{1}{1 - \frac{1}{1 - c o s h ^{2} ( x )}} = - \frac{1 - cosh ^{2} ( x )}{cosh ^{2} ( x )}

\frac{1}{1 - \frac{1}{1 - c o s h ^{2} ( x )}}

Simplificar

1 - \frac{1}{1 - cosh ^{2} ( x )}

en una fracción:

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

1 - \frac{1}{1 - cosh ^{2} ( x )}

Convertir a fracción:

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

= \frac{1 \cdot ( 1 - cosh ^{2} ( x ) )}{1 - cosh ^{2} ( x )} - \frac{1}{1 - cosh ^{2} ( x )}

Ya que los denominadores son iguales, combinar las fracciones:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot ( 1 - cosh ^{2} ( x ) ) - 1}{1 - cosh ^{2} ( x )}

1 \cdot (1 - cosh^{2} (x)) - 1 = - cosh^{2} (x)

1 \cdot (1 - cosh^{2} (x)) - 1

1 \cdot (1 - cosh^{2} (x)) = 1 - cosh^{2} (x)

1 \cdot (1 - cosh^{2} (x))

Multiplicar:

1 \cdot (1 - cosh^{2} (x)) = (1 - cosh^{2} (x))

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

Quitar los parentesis:

(a) = a

= 1 - cosh^{2} (x)

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

Agrupar términos semejantes

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

1 - 1 = 0

= - cosh^{2} (x)

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

Aplicar las propiedades de las fracciones:

\frac{- a}{b} = - \frac{a}{b}

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

= \frac{1}{- \frac{c o s h ^{2} ( x )}{- c o s h ^{2} ( x ) + 1}}

Aplicar las propiedades de las fracciones:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{1}{\frac{c o s h ^{2} ( x )}{1 - c o s h ^{2} ( x )}}

Aplicar las propiedades de las fracciones:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

\frac{1}{\frac{c o s h ^{2} ( x )}{1 - c o s h ^{2} ( x )}} = \frac{1 - cosh ^{2} ( x )}{cosh ^{2} ( x )}

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

= \frac{1}{1 - ( - \frac{- c o s h ^{2} ( x ) + 1}{c o s h ^{2} ( x )} )}

Aplicar la regla

- (- a) = a

= \frac{1}{1 + \frac{1 - c o s h ^{2} ( x )}{c o s h ^{2} ( x )}}

Simplificar

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

en una fracción:

\frac{1}{cosh ^{2} ( x )}

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

Convertir a fracción:

1 = \frac{1 cosh ^{2} ( x )}{cosh ^{2} ( x )}

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

Ya que los denominadores son iguales, combinar las fracciones:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

1 \cdot cosh^{2} (x) + 1 - cosh^{2} (x) = 1

1 \cdot cosh^{2} (x) + 1 - cosh^{2} (x)

Agrupar términos semejantes

= 1 \cdot cosh^{2} (x) - cosh^{2} (x) + 1

Sumar elementos similares:

1 \cdot cosh^{2} (x) - cosh^{2} (x) = 0

= 1

= \frac{1}{cosh ^{2} ( x )}

= \frac{1}{\frac{1}{c o s h ^{2} ( x )}}

Aplicar las propiedades de las fracciones:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{cosh ^{2} ( x )}{1}

Aplicar la regla

\frac{a}{1} = a

= cosh^{2} (x)

= cosh^{2} (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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