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verificar csc^2(x/2)=(2sec(x))/(sec(x)-1)

Solución

verificar

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

Solución

Verdadero

Pasos de solución

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

Sea:

u = \frac{x}{2}

csc^{2} (u) = \frac{2 sec ( 2 u )}{sec ( 2 u ) - 1}

Verificar

csc^{2} (u) = \frac{2 sec ( 2 u )}{sec ( 2 u ) - 1} :

Verdadero

csc^{2} (u) = \frac{2 sec ( 2 u )}{sec ( 2 u ) - 1}

Manipular el lado izquierdo

\frac{2 sec ( 2 u )}{sec ( 2 u ) - 1}

Expresar con seno, coseno

\frac{2 sec ( 2 u )}{- 1 + sec ( 2 u )}

Utilizar la identidad trigonométrica básica:

sec (x) = \frac{1}{cos ( x )}

= \frac{2 \cdot \frac{1}{c o s ( 2 u )}}{- 1 + \frac{1}{c o s ( 2 u )}}

Simplificar

\frac{2 \cdot \frac{1}{c o s ( 2 u )}}{- 1 + \frac{1}{c o s ( 2 u )}} : \frac{2}{- cos ( 2 u ) + 1}

\frac{2 \cdot \frac{1}{c o s ( 2 u )}}{- 1 + \frac{1}{c o s ( 2 u )}}

Simplificar

- 1 + \frac{1}{cos ( 2 u )}

en una fracción:

\frac{- cos ( 2 u ) + 1}{cos ( 2 u )}

- 1 + \frac{1}{cos ( 2 u )}

Convertir a fracción:

1 = \frac{1 cos ( 2 u )}{cos ( 2 u )}

= - \frac{1 \cdot cos ( 2 u )}{cos ( 2 u )} + \frac{1}{cos ( 2 u )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- 1 \cdot cos ( 2 u ) + 1}{cos ( 2 u )}

Multiplicar:

1 \cdot cos (2 u) = cos (2 u)

= \frac{- cos ( 2 u ) + 1}{cos ( 2 u )}

= \frac{2 \cdot \frac{1}{c o s ( 2 u )}}{\frac{- c o s ( 2 u ) + 1}{c o s ( 2 u )}}

Multiplicar

2 \cdot \frac{1}{cos ( 2 u )} : \frac{2}{cos ( 2 u )}

2 \cdot \frac{1}{cos ( 2 u )}

Multiplicar fracciones:

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

= \frac{1 \cdot 2}{cos ( 2 u )}

Multiplicar los numeros:

1 \cdot 2 = 2

= \frac{2}{cos ( 2 u )}

= \frac{\frac{2}{c o s ( 2 u )}}{\frac{- c o s ( 2 u ) + 1}{c o s ( 2 u )}}

Dividir fracciones:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

= \frac{2 cos ( 2 u )}{cos ( 2 u ) ( - cos ( 2 u ) + 1 )}

Eliminar los terminos comunes:

cos (2 u)

= \frac{2}{- cos ( 2 u ) + 1}

= \frac{2}{1 - cos ( 2 u )}

= \frac{2}{1 - cos ( 2 u )}

Re-escribir usando identidades trigonométricas

\frac{2}{1 - cos ( 2 u )}

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Simplificar

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

\frac{2}{1 - ( 1 - 2 sin ^{2} ( u ) )}

Expandir

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

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

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

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

Poner los parentesis

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

Aplicar las reglas de los signos

- (- a) = a, - (a) = - a

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

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

1 - 1 = 0

= 2 sin^{2} (u)

= \frac{2}{2 sin ^{2} ( u )}

Dividir:

\frac{2}{2} = 1

= \frac{1}{sin ^{2} ( u )}

= \frac{1}{sin ^{2} ( u )}

= \frac{1}{sin ^{2} ( u )}

Re-escribir usando identidades trigonométricas

Utilizar la identidad trigonométrica básica:

sin (x) = \frac{1}{csc ( x )}

\frac{1}{( \frac{1}{c s c ( u )} ) ^{2}}

Simplificar

\frac{1}{( \frac{1}{c s c ( u )} ) ^{2}}

(\frac{1}{csc ( u )})^{2} = \frac{1}{csc ^{2} ( u )}

(\frac{1}{csc ( u )})^{2}

Aplicar las leyes de los exponentes:

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

= \frac{1 ^{2}}{csc ^{2} ( u )}

Aplicar la regla

1^{a} = 1

1^{2} = 1

= \frac{1}{csc ^{2} ( u )}

= \frac{1}{\frac{1}{c s c ^{2} ( u )}}

Aplicar las propiedades de las fracciones:

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

= \frac{csc ^{2} ( u )}{1}

Aplicar la regla

\frac{a}{1} = a

= csc^{2} (u)

csc^{2} (u)

csc^{2} (u)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

Por lo tanto

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

Ejemplos populares

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verificar 1=(sec(u)+tan(u))/(sec(u)+tan(u))verificar

1 = \frac{sec ( u ) + tan ( u )}{sec ( u ) + tan ( u )}

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

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