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

Solución

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

Solución

Sin soluci \overset{o}{ˊ} n para x \in R

Pasos de solución

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

Restar

cos^{2} (x)

de ambos lados

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

Elevar al cuadrado ambos lados

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

Restar

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

de ambos lados

sin^{4} (x) - 4 + 4 cos^{2} (x) - cos^{4} (x) = 0

Aplicar regla de los exponentes:

a^{b} = a^{2} a^{b - 2}

- 4 - cos^{4} (x) + 4 cos^{2} (x) + sin^{2} (x) sin^{2} (x) = 0

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

- 4 - cos^{4} (x) + 4 cos^{2} (x) + (1 - cos^{2} (x)) (1 - cos^{2} (x)) : 2 cos^{2} (x) - 3

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

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

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

Aplicar las leyes de los exponentes:

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

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

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

Sumar:

1 + 1 = 2

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

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

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

Aplicar la formula del binomio al cuadrado:

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

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

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

Simplificar

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

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

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

Multiplicar los numeros:

2 \cdot 1 = 2

= 2 cos^{2} (x)

(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)

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

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

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

Simplificar

- 4 - cos^{4} (x) + 4 cos^{2} (x) + 1 - 2 cos^{2} (x) + cos^{4} (x) : 2 cos^{2} (x) - 3

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

Agrupar términos semejantes

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

Sumar elementos similares:

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

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

Sumar elementos similares:

- cos^{4} (x) + cos^{4} (x) = 0

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

Sumar/restar lo siguiente:

- 4 + 1 = - 3

= 2 cos^{2} (x) - 3

= 2 cos^{2} (x) - 3

= 2 cos^{2} (x) - 3

- 3 + 2 cos^{2} (x) = 0

Usando el método de sustitución

- 3 + 2 cos^{2} (x) = 0

Sea:

cos (x) = u

- 3 + 2 u^{2} = 0

- 3 + 2 u^{2} = 0 : u = \frac{3}{2}, u = - \frac{3}{2}

- 3 + 2 u^{2} = 0

Mover

3

al lado derecho

- 3 + 2 u^{2} = 0

Sumar

3

a ambos lados

- 3 + 2 u^{2} + 3 = 0 + 3

Simplificar

2 u^{2} = 3

2 u^{2} = 3

Dividir ambos lados entre

2

2 u^{2} = 3

Dividir ambos lados entre

2

\frac{2 u ^{2}}{2} = \frac{3}{2}

Simplificar

u^{2} = \frac{3}{2}

u^{2} = \frac{3}{2}

Para

x^{2} = f (a)

las soluciones son

x = f (a), - f (a)

u = \frac{3}{2}, u = - \frac{3}{2}

Sustituir en la ecuación

u = cos (x)

cos (x) = \frac{3}{2}, cos (x) = - \frac{3}{2}

cos (x) = \frac{3}{2}, cos (x) = - \frac{3}{2}

cos (x) = \frac{3}{2} :

Sin solución

cos (x) = \frac{3}{2}

- 1 \leq cos (x) \leq 1

Sin soluci \overset{o}{ˊ} n

cos (x) = - \frac{3}{2} :

Sin solución

cos (x) = - \frac{3}{2}

- 1 \leq cos (x) \leq 1

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

Sin soluci \overset{o}{ˊ} n

Verificar las soluciones sustituyendo en la ecuación original

Verificar las soluciones sustituyéndolas en

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

Quitar las que no concuerden con la ecuación.

Sin soluci \overset{o}{ˊ} n para x \in R

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