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

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

csc^{2} (x) (sec^{2} (x) - 1)

Expresar con seno, coseno

(- 1 + sec^{2} (x)) csc^{2} (x)

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

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

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

Aplicar las leyes de los exponentes:

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

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

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

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

Aplicar las leyes de los exponentes:

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

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

Multiplicar fracciones:

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

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

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

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

Multiplicar:

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

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

Quitar los parentesis:

(- a) = - a

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

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

Simplificar

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

en una fracción:

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

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

Convertir a fracción:

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

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

Multiplicar:

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

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

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

Aplicar las propiedades de las fracciones:

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

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Eliminar los terminos comunes:

sin^{2} (x)

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

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

sin^{2} (0) = 4 (cos (- 0) - 1)

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

cos (\frac{π}{2} - x) sec (x) = tan (x)

cos (- x) + sin (- x) = cos (x) - sin (x)

\frac{1 - cos ( 2 θ )}{cos ( 2 θ ) + 1} = tan^{2} (θ)