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cos^4(2x)-sin^4(2x)=1

Solución

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

Solución

x = πn, x = \frac{π}{2} + πn

Grados

x = 0^{\circ} + 18 0^{\circ} n, x = 9 0^{\circ} + 18 0^{\circ} n

Pasos de solución

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

Restar

1

de ambos lados

cos^{4} (2 x) - sin^{4} (2 x) - 1 = 0

Aplicar regla de los exponentes:

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

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

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

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

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

Aplicar las leyes de los exponentes:

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

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

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

Sumar:

1 + 1 = 2

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

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

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

Aplicar la formula del binomio al cuadrado:

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

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

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

Simplificar

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

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

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

Multiplicar los numeros:

2 \cdot 1 = 2

= 2 sin^{2} (2 x)

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

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

Aplicar las leyes de los exponentes:

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

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

Multiplicar los numeros:

2 \cdot 2 = 4

= sin^{4} (2 x)

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

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

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

Simplificar

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

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

Agrupar términos semejantes

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

Sumar elementos similares:

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

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

- 1 + 1 = 0

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

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

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

- 2 sin^{2} (2 x) = 0

Dividir ambos lados entre

- 2

- 2 sin^{2} (2 x) = 0

Dividir ambos lados entre

- 2

- 2 sin^{2} (2 x) = 0

Dividir ambos lados entre

- 2

\frac{- 2 sin ^{2} ( 2 x )}{- 2} = \frac{0}{- 2}

Simplificar

sin^{2} (2 x) = 0

sin^{2} (2 x) = 0

Aplicar la regla

x^{n} = 0 \Rightarrow x = 0

sin (2 x) = 0

Soluciones generales para

sin (2 x) = 0

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

2 x = 0 + 2 πn, 2 x = π + 2 πn

2 x = 0 + 2 πn, 2 x = π + 2 πn

Resolver

2 x = 0 + 2 πn : x = πn

2 x = 0 + 2 πn

0 + 2 πn = 2 πn

2 x = 2 πn

Dividir ambos lados entre

2

2 x = 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{2 πn}{2}

Simplificar

x = πn

x = πn

Resolver

2 x = π + 2 πn : x = \frac{π}{2} + πn

2 x = π + 2 πn

Dividir ambos lados entre

2

2 x = π + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{π}{2} + \frac{2 πn}{2}

Simplificar

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

x = πn, x = \frac{π}{2} + πn

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\frac{125}{sin ( 11 0 ^{\circ} )} = \frac{100}{sin ( b )}

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tan (\frac{2 x}{3}) - 3 = 0