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verificar 4/(cos^2(x))-5=4tan^2(x)-1

Solución

verificar

\frac{4}{cos ^{2} ( x )} - 5 = 4 tan^{2} (x) - 1

Verdadero

Pasos de solución

\frac{4}{cos ^{2} ( x )} - 5 = 4 tan^{2} (x) - 1

Manipular el lado izquierdo

4 tan^{2} (x) - 1

Expresar con seno, coseno

- 1 + 4 tan^{2} (x)

Utilizar la identidad trigonométrica básica:

tan (x) = \frac{sin ( x )}{cos ( x )}

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

Simplificar

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

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

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

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

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

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

Aplicar las leyes de los exponentes:

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

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

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

Multiplicar fracciones:

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

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

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

Multiplicar:

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

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Expandir

- cos^{2} (x) + 4 (1 - cos^{2} (x)) : - 5 cos^{2} (x) + 4

- cos^{2} (x) + 4 (1 - cos^{2} (x))

Expandir

4 (1 - cos^{2} (x)) : 4 - 4 cos^{2} (x)

4 (1 - cos^{2} (x))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = 4, b = 1, c = cos^{2} (x)

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

Multiplicar los numeros:

4 \cdot 1 = 4

= 4 - 4 cos^{2} (x)

= - cos^{2} (x) + 4 - 4 cos^{2} (x)

Simplificar

- cos^{2} (x) + 4 - 4 cos^{2} (x) : - 5 cos^{2} (x) + 4

- cos^{2} (x) + 4 - 4 cos^{2} (x)

Agrupar términos semejantes

= - cos^{2} (x) - 4 cos^{2} (x) + 4

Sumar elementos similares:

- cos^{2} (x) - 4 cos^{2} (x) = - 5 cos^{2} (x)

= - 5 cos^{2} (x) + 4

= - 5 cos^{2} (x) + 4

= \frac{- 5 cos ^{2} ( x ) + 4}{cos ^{2} ( x )}

= \frac{- 5 cos ^{2} ( x ) + 4}{cos ^{2} ( x )}

Expandir

\frac{4 - 5 cos ^{2} ( x )}{cos ^{2} ( x )} : \frac{4}{cos ^{2} ( x )} - 5

\frac{4 - 5 cos ^{2} ( x )}{cos ^{2} ( x )}

Aplicar las propiedades de las fracciones:

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

\frac{4 - 5 cos ^{2} ( x )}{cos ^{2} ( x )} = \frac{4}{cos ^{2} ( x )} - \frac{5 cos ^{2} ( x )}{cos ^{2} ( x )}

= \frac{4}{cos ^{2} ( x )} - \frac{5 cos ^{2} ( x )}{cos ^{2} ( x )}

Cancelar

\frac{5 cos ^{2} ( x )}{cos ^{2} ( x )} : 5

\frac{5 cos ^{2} ( x )}{cos ^{2} ( x )}

Eliminar los terminos comunes:

cos^{2} (x)

= 5

= \frac{4}{cos ^{2} ( x )} - 5

= - 5 + \frac{4}{cos ^{2} ( x )}

= \frac{4}{cos ^{2} ( x )} - 5

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero