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verificar (2sin(x)+cos(x))^2+(2cos(x)-sin(x))^2=5

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

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

Factorizar

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

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

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

Aplicar la formula del binomio al cuadrado:

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

a = - sin (x), b = 2 cos (x)

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

Simplificar

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

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

Quitar los parentesis:

(- a) = - a

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

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

(- sin (x))^{2}

Aplicar las leyes de los exponentes:

(- a)^{n} = a^{n},

n

es par

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

= sin^{2} (x)

2 sin (x) \cdot 2 cos (x) = 4 sin (x) cos (x)

2 sin (x) \cdot 2 cos (x)

Multiplicar los numeros:

2 \cdot 2 = 4

= 4 sin (x) cos (x)

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

(2 cos (x))^{2}

Aplicar las leyes de los exponentes:

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

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

2^{2} = 4

= 4 cos^{2} (x)

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

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

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

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

Aplicar la formula del binomio al cuadrado:

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

a = cos (x), b = 2 sin (x)

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

Simplificar

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

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

2 cos (x) \cdot 2 sin (x) = 4 cos (x) sin (x)

2 cos (x) \cdot 2 sin (x)

Multiplicar los numeros:

2 \cdot 2 = 4

= 4 cos (x) sin (x)

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

(2 sin (x))^{2}

Aplicar las leyes de los exponentes:

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

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

2^{2} = 4

= 4 sin^{2} (x)

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

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

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

Simplificar

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

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

Sumar elementos similares:

- 4 sin (x) cos (x) + 4 cos (x) sin (x) = 0

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

Sumar elementos similares:

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

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

Sumar elementos similares:

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

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

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

Factorizar el termino común

5

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

= (cos^{2} (x) + sin^{2} (x)) \cdot 5

Re-escribir usando identidades trigonométricas

(cos^{2} (x) + sin^{2} (x)) \cdot 5

Utilizar la identidad pitagórica:

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

= 1 \cdot 5

Simplificar

= 5

= 5

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

cos (a) (sec (a) - cos (a)) = sin^{2} (a)

tan (x) cos (x) csc (x) = 1

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

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

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