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verificar csch^2(x)=coth^2(x)-1

Solución

verificar

csch^{2} (x) = coth^{2} (x) - 1

Verdadero

Pasos de solución

csch^{2} (x) = coth^{2} (x) - 1

Manipular el lado izquierdo

coth^{2} (x) - 1

Re-escribir usando identidades trigonométricas

coth^{2} (x) - 1

Utilizar la identidad hiperbólica:

coth (x) = \frac{cosh ( x )}{sinh ( x )}

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

Reescribir

1

como

\frac{sinh ^{2} ( x )}{sinh ^{2} ( x )}

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

Simplificar

(\frac{cosh ( x )}{sinh ( x )})^{2} - \frac{sinh ^{2} ( x )}{sinh ^{2} ( x )} : \frac{cosh ^{2} ( x ) - sinh ^{2} ( x )}{sinh ^{2} ( x )}

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

Aplicar la regla

\frac{a}{a} = 1

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

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

Aplicar las leyes de los exponentes:

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

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

Multiplicar:

1 \cdot sinh^{2} (x) = sinh^{2} (x)

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad hiperbólica:

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

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

Utilizar la identidad hiperbólica:

sinh (x) = \frac{1}{csch ( x )}

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

Simplificar

\frac{1}{( \frac{1}{csch ( x )} ) ^{2}} : csch^{2} (x)

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

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

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

Aplicar las leyes de los exponentes:

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

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

Aplicar las propiedades de las fracciones:

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

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

Aplicar la regla

\frac{a}{1} = a

= csch^{2} (x)

= csch^{2} (x)

= csch^{2} (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero