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verificar 1-2sin^2(2x)=8cos^4(x)-8cos^2(x)+1

Solución

verificar

1 - 2 sin^{2} (2 x) = 8 cos^{4} (x) - 8 cos^{2} (x) + 1

Solución

Verdadero

Pasos de solución

1 - 2 sin^{2} (2 x) = 8 cos^{4} (x) - 8 cos^{2} (x) + 1

Manipular el lado derecho

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

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

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

Expandir

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

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

Poner los parentesis utilizando:

a (b - c) = ab - a c

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

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

Aplicar las reglas de los signos

- (- a) = a

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

Multiplicar los numeros:

2 \cdot 1 = 2

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

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

Restar:

1 - 2 = - 1

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

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

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Simplificar

2 (2 cos^{2} (x) - 1)^{2} - 1 : 8 cos^{4} (x) - 8 cos^{2} (x) + 1

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

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

Aplicar la formula del binomio al cuadrado:

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

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

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

Simplificar

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

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

(2 cos^{2} (x))^{2}

Aplicar las leyes de los exponentes:

(a \cdot b)^{n} = a^{n} b^{n}

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

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

Aplicar las leyes de los exponentes:

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

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

Multiplicar los numeros:

2 \cdot 2 = 4

= cos^{4} (x)

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

2^{2} = 4

= 4 cos^{4} (x)

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

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

Multiplicar los numeros:

2 \cdot 2 \cdot 1 = 4

= 4 cos^{2} (x)

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

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

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

Expandir

2 (4 cos^{4} (x) - 4 cos^{2} (x) + 1) : 8 cos^{4} (x) - 8 cos^{2} (x) + 2

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

Aplicar la siguiente regla de productos notables

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

Aplicar las reglas de los signos

+ (- a) = - a

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

Simplificar

2 \cdot 4 cos^{4} (x) - 2 \cdot 4 cos^{2} (x) + 2 \cdot 1 : 8 cos^{4} (x) - 8 cos^{2} (x) + 2

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

Multiplicar los numeros:

2 \cdot 4 = 8

= 8 cos^{4} (x) - 8 cos^{2} (x) + 2 \cdot 1

Multiplicar los numeros:

2 \cdot 1 = 2

= 8 cos^{4} (x) - 8 cos^{2} (x) + 2

= 8 cos^{4} (x) - 8 cos^{2} (x) + 2

= 8 cos^{4} (x) - 8 cos^{2} (x) + 2 - 1

Sumar/restar lo siguiente:

2 - 1 = 1

= 8 cos^{4} (x) - 8 cos^{2} (x) + 1

= 8 cos^{4} (x) - 8 cos^{2} (x) + 1

= 8 cos^{4} (x) - 8 cos^{2} (x) + 1

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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