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

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

cos (4 a)

Re-escribir usando identidades trigonométricas

cos (4 a)

= cos (2 \cdot 2 a)

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Simplificar

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

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

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

Aplicar la formula del binomio al cuadrado:

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

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

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

Simplificar

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

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

(2 cos^{2} (a))^{2}

Aplicar las leyes de los exponentes:

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

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

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

Aplicar las leyes de los exponentes:

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

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

Multiplicar los numeros:

2 \cdot 2 = 4

= cos^{4} (a)

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

2^{2} = 4

= 4 cos^{4} (a)

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

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

Multiplicar los numeros:

2 \cdot 2 \cdot 1 = 4

= 4 cos^{2} (a)

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

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

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

Expandir

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

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

Aplicar la siguiente regla de productos notables

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

Aplicar las reglas de los signos

+ (- a) = - a

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

Simplificar

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

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

Multiplicar los numeros:

2 \cdot 4 = 8

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

Multiplicar los numeros:

2 \cdot 1 = 2

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

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

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

Sumar/restar lo siguiente:

2 - 1 = 1

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

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

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

2 cos^{2} (x) + cos (x) = 0

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

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

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

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