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Popular Trigonometría >

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

Solución

verificar

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

Solución

Verdadero

Pasos de solución

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

Manipular el lado derecho

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Simplificar

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

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

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

Aplicar la formula del binomio al cuadrado:

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

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

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

Simplificar

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

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

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

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

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)

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

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

Aplicar las leyes de los exponentes:

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

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

Multiplicar los numeros:

2 \cdot 2 = 4

= sin^{4} (x)

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

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

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

Sumar elementos similares:

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

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

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

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

Factorizar

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

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

Factorizar la expresión

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

Reescribir en la forma

a x^{2} + b x y + c y^{2}

dejar

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

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

Definición

Factores de

1 : 1

1

Divisores (factores)

Encontrar los factores primos de

1 :

Agregar 1

1

Divisores de

1

1

Por cada dos factores tales que

u * v = 1,

revisar si

u + v = 2

Revisar

u = 1, v = 1 : u * v = 1, u + v = 2 \Rightarrow

Verdadero

u = 1, v = 1

Agrupar en

(a x^{2} + ux y^{2}) + (v x^{2} y + c y^{2})

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

Reiniciar

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

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

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

Factorizar

sin^{2} (x)

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

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

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

Aplicar las leyes de los exponentes:

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

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

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

Factorizar el termino común

sin^{2} (x)

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

Factorizar

cos^{2} (x)

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

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

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

Aplicar las leyes de los exponentes:

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

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

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

Factorizar el termino común

cos^{2} (x)

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

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

Factorizar el termino común

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

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

Simplificar

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

= 1^{2}

Aplicar la regla

1^{a} = 1

= 1

= 1

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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