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verificar 1-tan^4(x)=2sec^2(x)-sec^4(x)

Solución

verificar

1 - tan^{4} (x) = 2 sec^{2} (x) - sec^{4} (x)

Verdadero

Pasos de solución

1 - tan^{4} (x) = 2 sec^{2} (x) - sec^{4} (x)

Manipular el lado izquierdo

2 sec^{2} (x) - sec^{4} (x)

Factorizar

- sec^{4} (x) + 2 sec^{2} (x) : sec^{2} (x) (- sec^{2} (x) + 2)

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

Aplicar las leyes de los exponentes:

a^{b + c} = a^{b} a^{c}

sec^{4} (x) = sec^{2} (x) sec^{2} (x)

= - sec^{2} (x) sec^{2} (x) + 2 sec^{2} (x)

Factorizar el termino común

sec^{2} (x)

= sec^{2} (x) (- sec^{2} (x) + 2)

= (2 - sec^{2} (x)) sec^{2} (x)

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

sec^{2} (x) = tan^{2} (x) + 1

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

Expandir

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

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

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

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

Poner los parentesis

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

Aplicar las reglas de los signos

+ (- a) = - a

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

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

Simplificar

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

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

Agrupar términos semejantes

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

Sumar/restar lo siguiente:

2 - 1 = 1

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

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

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

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

Expandir

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

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

Aplicar la siguiente regla para binomios al cuadrado:

(a + b) (a - b) = a^{2} - b^{2}

a = 1, b = tan^{2} (x)

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

Simplificar

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

(tan^{2} (x))^{2} = tan^{4} (x)

(tan^{2} (x))^{2}

Aplicar las leyes de los exponentes:

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

= tan^{2 \cdot 2} (x)

Multiplicar los numeros:

2 \cdot 2 = 4

= tan^{4} (x)

= 1 - tan^{4} (x)

= 1 - tan^{4} (x)

= 1 - tan^{4} (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

sin (π) = sin (- π)

csc^{2} (x) + 2 cot (x) = (1 + cot (x))^{2}

csc (t) cot (t) = \frac{1 + cot ^{2} ( t )}{sec ( t )}

csc (a) sec (a) = 2 csc (2 a)

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