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Popular Trigonometria >

provar sin^6(a)+cos^6(a)=1-3sin^2(a)*cos^2(a)

Solução

provar

sin^{6} (a) + cos^{6} (a) = 1 - 3 sin^{2} (a) \cdot cos^{2} (a)

Solução

Verdadeiro

Passos da solução

sin^{6} (a) + cos^{6} (a) = 1 - 3 sin^{2} (a) cos^{2} (a)

Manipular o lado direito

sin^{6} (a) + cos^{6} (a)

Fatorar

cos^{6} (a) + sin^{6} (a) : (cos^{2} (a) + sin^{2} (a)) (cos^{4} (a) - sin^{2} (a) cos^{2} (a) + sin^{4} (a))

cos^{6} (a) + sin^{6} (a)

Reescrever

cos^{6} (a) + sin^{6} (a)

como

(cos^{2} (a))^{3} + (sin^{2} (a))^{3}

cos^{6} (a) + sin^{6} (a)

Aplicar as propriedades dos expoentes:

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

sin^{6} (a) = (sin^{2} (a))^{3}

= cos^{6} (a) + (sin^{2} (a))^{3}

Aplicar as propriedades dos expoentes:

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

cos^{6} (a) = (cos^{2} (a))^{3}

= (cos^{2} (a))^{3} + (sin^{2} (a))^{3}

= (cos^{2} (a))^{3} + (sin^{2} (a))^{3}

Aplicar a regra da soma de cubos:

x^{3} + y^{3} = (x + y) (x^{2} - x y + y^{2})

(cos^{2} (a))^{3} + (sin^{2} (a))^{3} = (cos^{2} (a) + sin^{2} (a)) (cos^{4} (a) - sin^{2} (a) cos^{2} (a) + sin^{4} (a))

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

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

Reeecreva usando identidades trigonométricas

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

Utilizar a identidade trigonométrica pitagórica:

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

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

Simplificar

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

Utilizar a identidade trigonométrica do arco duplo:

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

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

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

Simplificar

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

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

Expandir

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

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

Colocar os parênteses utilizando:

a (b - c) = ab - a c

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

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

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

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

Aplicar as propriedades dos expoentes:

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

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

= cos^{2 + 2} (a)

Somar:

2 + 2 = 4

= cos^{4} (a)

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

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

Somar elementos similares:

cos^{4} (a) - cos^{4} (a) = 0

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

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

Utilizar a identidade trigonométrica do arco duplo:

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

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

Utilizar a identidade trigonométrica pitagórica:

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

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

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

Simplificar

sin^{4} (a) + (1 - sin^{2} (a)) (1 - 2 sin^{2} (a)) : 3 sin^{4} (a) - 3 sin^{2} (a) + 1

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

Expandir

(1 - sin^{2} (a)) (1 - 2 sin^{2} (a)) : 1 - 3 sin^{2} (a) + 2 sin^{4} (a)

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

Aplique o método FOIL:

(a + b) (c + d) = a c + a d + b c + b d

a = 1, b = - sin^{2} (a), c = 1, d = - 2 sin^{2} (a)

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

Aplicar as regras dos sinais

+ (- a) = - a, (- a) (- b) = ab

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

Simplificar

1 \cdot 1 - 1 \cdot 2 sin^{2} (a) - 1 \cdot sin^{2} (a) + 2 sin^{2} (a) sin^{2} (a) : 1 - 3 sin^{2} (a) + 2 sin^{4} (a)

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

1 \cdot 1 = 1

1 \cdot 1

Multiplicar os números:

1 \cdot 1 = 1

= 1

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

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

Multiplicar os números:

1 \cdot 2 = 2

= 2 sin^{2} (a)

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

1 \cdot sin^{2} (a)

Multiplicar:

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

= sin^{2} (a)

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

2 sin^{2} (a) sin^{2} (a)

Aplicar as propriedades dos expoentes:

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

sin^{2} (a) sin^{2} (a) = sin^{2 + 2} (a)

= 2 sin^{2 + 2} (a)

Somar:

2 + 2 = 4

= 2 sin^{4} (a)

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

Somar elementos similares:

- 2 sin^{2} (a) - sin^{2} (a) = - 3 sin^{2} (a)

= 1 - 3 sin^{2} (a) + 2 sin^{4} (a)

= 1 - 3 sin^{2} (a) + 2 sin^{4} (a)

= sin^{4} (a) + 1 - 3 sin^{2} (a) + 2 sin^{4} (a)

Simplificar

sin^{4} (a) + 1 - 3 sin^{2} (a) + 2 sin^{4} (a) : 3 sin^{4} (a) - 3 sin^{2} (a) + 1

sin^{4} (a) + 1 - 3 sin^{2} (a) + 2 sin^{4} (a)

Agrupar termos semelhantes

= sin^{4} (a) - 3 sin^{2} (a) + 2 sin^{4} (a) + 1

Somar elementos similares:

sin^{4} (a) + 2 sin^{4} (a) = 3 sin^{4} (a)

= 3 sin^{4} (a) - 3 sin^{2} (a) + 1

= 3 sin^{4} (a) - 3 sin^{2} (a) + 1

= 3 sin^{4} (a) - 3 sin^{2} (a) + 1

= 3 sin^{4} (a) - 3 sin^{2} (a) + 1

Fatorar

- 3 sin^{2} (a) + 3 sin^{4} (a) : 3 sin^{2} (a) (sin (a) + 1) (sin (a) - 1)

- 3 sin^{2} (a) + 3 sin^{4} (a)

Fatorar o termo comum

3 sin^{2} (a) : 3 sin^{2} (a) (sin^{2} (a) - 1)

3 sin^{4} (a) - 3 sin^{2} (a)

Aplicar as propriedades dos expoentes:

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

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

= 3 sin^{2} (a) sin^{2} (a) - 3 sin^{2} (a)

Fatorar o termo comum

3 sin^{2} (a)

= 3 sin^{2} (a) (sin^{2} (a) - 1)

= 3 sin^{2} (a) (sin^{2} (a) - 1)

Fatorar

sin^{2} (a) - 1 : (sin (a) + 1) (sin (a) - 1)

sin^{2} (a) - 1

Reescrever

1

como

1^{2}

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

Aplicar a regra da diferença de quadrados:

x^{2} - y^{2} = (x + y) (x - y)

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

= (sin (a) + 1) (sin (a) - 1)

= 3 sin^{2} (a) (sin (a) + 1) (sin (a) - 1)

= 1 + (- 1 + sin (a)) (1 + sin (a)) \cdot 3 sin^{2} (a)

Reeecreva usando identidades trigonométricas

1 + (- 1 + sin (a)) (1 + sin (a)) \cdot 3 sin^{2} (a)

Expandir

(sin (a) + 1) (sin (a) - 1) : sin^{2} (a) - 1

(sin (a) + 1) (sin (a) - 1)

Aplicar a regra da diferença de quadrados:

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

a = sin (a), b = 1

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

Aplicar a regra

1^{a} = 1

1^{2} = 1

= sin^{2} (a) - 1

= 1 + 3 sin^{2} (a) (sin^{2} (a) - 1)

Utilizar a identidade trigonométrica pitagórica:

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

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

= 1 + 3 sin^{2} (a) (- cos^{2} (a))

Simplificar

= 1 - 3 sin^{2} (a) cos^{2} (a)

= 1 - 3 sin^{2} (a) cos^{2} (a)

Demonstramos que os dois lados podem adquirir a mesma forma

\Rightarrow Verdadeiro

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