Trigonometria Exemplos

$\cos^{2} (3 x) - \sin^{2} (3 x)$

Etapa 1

Como os dois termos são quadrados perfeitos, fatore usando a fórmula da diferença de quadrados, $a^{2} - b^{2} = (a + b) (a - b)$ em que $a = \cos (3 x)$ e $b = \sin (3 x)$ .

$(\cos (3 x) + \sin (3 x)) (\cos (3 x) - \sin (3 x))$

Etapa 2

Simplifique os termos.

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Etapa 2.1

Simplifique cada termo.

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Etapa 2.1.1

Use a fórmula do arco triplo para transformar $\cos (3 x)$ em $4 \cos^{3} (x) - 3 \cos (x)$ .

$(4 \cos^{3} (x) - 3 \cos (x) + \sin (3 x)) (\cos (3 x) - \sin (3 x))$

Etapa 2.1.2

Aplique a fórmula do arco triplo do seno.

$(4 \cos^{3} (x) - 3 \cos (x) - 4 \sin^{3} (x) + 3 \sin (x)) (\cos (3 x) - \sin (3 x))$

Etapa 2.2

Simplifique cada termo.

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Etapa 2.2.1

Use a fórmula do arco triplo para transformar $\cos (3 x)$ em $4 \cos^{3} (x) - 3 \cos (x)$ .

$(4 \cos^{3} (x) - 3 \cos (x) - 4 \sin^{3} (x) + 3 \sin (x)) (4 \cos^{3} (x) - 3 \cos (x) - \sin (3 x))$

Etapa 2.2.2

Aplique a fórmula do arco triplo do seno.

$(4 \cos^{3} (x) - 3 \cos (x) - 4 \sin^{3} (x) + 3 \sin (x)) (4 \cos^{3} (x) - 3 \cos (x) - (- 4 \sin^{3} (x) + 3 \sin (x)))$

Etapa 2.2.3

Aplique a propriedade distributiva.

$(4 \cos^{3} (x) - 3 \cos (x) - 4 \sin^{3} (x) + 3 \sin (x)) (4 \cos^{3} (x) - 3 \cos (x) - (- 4 \sin^{3} (x)) - (3 \sin (x)))$

Etapa 2.2.4

Multiplique $- 4$ por $- 1$ .

$(4 \cos^{3} (x) - 3 \cos (x) - 4 \sin^{3} (x) + 3 \sin (x)) (4 \cos^{3} (x) - 3 \cos (x) + 4 \sin^{3} (x) - (3 \sin (x)))$

Etapa 2.2.5

Multiplique $3$ por $- 1$ .

$(4 \cos^{3} (x) - 3 \cos (x) - 4 \sin^{3} (x) + 3 \sin (x)) (4 \cos^{3} (x) - 3 \cos (x) + 4 \sin^{3} (x) - 3 \sin (x))$

Etapa 3

Expanda $(4 \cos^{3} (x) - 3 \cos (x) - 4 \sin^{3} (x) + 3 \sin (x)) (4 \cos^{3} (x) - 3 \cos (x) + 4 \sin^{3} (x) - 3 \sin (x))$ multiplicando cada termo na primeira expressão por cada um dos termos na segunda expressão.

$4 \cos^{3} (x) (4 \cos^{3} (x)) + 4 \cos^{3} (x) (- 3 \cos (x)) + 4 \cos^{3} (x) (4 \sin^{3} (x)) + 4 \cos^{3} (x) (- 3 \sin (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (4 \cos^{3} (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (4 \cos^{3} (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4

Simplifique os termos.

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Etapa 4.1

Combine os termos opostos em $4 \cos^{3} (x) (4 \cos^{3} (x)) + 4 \cos^{3} (x) (- 3 \cos (x)) + 4 \cos^{3} (x) (4 \sin^{3} (x)) + 4 \cos^{3} (x) (- 3 \sin (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (4 \cos^{3} (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (4 \cos^{3} (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$ .

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Etapa 4.1.1

Reorganize os fatores nos termos $4 \cos^{3} (x) (4 \sin^{3} (x))$ e $- 4 \sin^{3} (x) (4 \cos^{3} (x))$ .

$4 \cos^{3} (x) (4 \cos^{3} (x)) + 4 \cos^{3} (x) (- 3 \cos (x)) + 4 \cdot 4 \cos^{3} (x) \sin^{3} (x) + 4 \cos^{3} (x) (- 3 \sin (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \cdot 4 \cos^{3} (x) \sin^{3} (x) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (4 \cos^{3} (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.1.2

Subtraia $4 \cdot 4 \cos^{3} (x) \sin^{3} (x)$ de $4 \cdot 4 \cos^{3} (x) \sin^{3} (x)$ .

$4 \cos^{3} (x) (4 \cos^{3} (x)) + 4 \cos^{3} (x) (- 3 \cos (x)) + 4 \cos^{3} (x) (- 3 \sin (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) + 0 - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (4 \cos^{3} (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.1.3

Some $4 \cos^{3} (x) (4 \cos^{3} (x)) + 4 \cos^{3} (x) (- 3 \cos (x)) + 4 \cos^{3} (x) (- 3 \sin (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x))$ e $0$ .

$4 \cos^{3} (x) (4 \cos^{3} (x)) + 4 \cos^{3} (x) (- 3 \cos (x)) + 4 \cos^{3} (x) (- 3 \sin (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (4 \cos^{3} (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.1.4

Reorganize os fatores nos termos $4 \cos^{3} (x) (- 3 \sin (x))$ e $3 \sin (x) (4 \cos^{3} (x))$ .

$4 \cos^{3} (x) (4 \cos^{3} (x)) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cdot 4 \cos^{3} (x) \sin (x) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \cdot 4 \cos^{3} (x) \sin (x) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.1.5

Some $- 3 \cdot 4 \cos^{3} (x) \sin (x)$ e $3 \cdot 4 \cos^{3} (x) \sin (x)$ .

$4 \cos^{3} (x) (4 \cos^{3} (x)) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 0 + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.1.6

Some $4 \cos^{3} (x) (4 \cos^{3} (x)) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x))$ e $0$ .

$4 \cos^{3} (x) (4 \cos^{3} (x)) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2

Simplifique cada termo.

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Etapa 4.2.1

Reescreva usando a propriedade comutativa da multiplicação.

$4 \cdot 4 \cos^{3} (x) \cos^{3} (x) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.2

Multiplique $\cos^{3} (x)$ por $\cos^{3} (x)$ somando os expoentes.

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Etapa 4.2.2.1

Mova $\cos^{3} (x)$ .

$4 \cdot 4 (\cos^{3} (x) \cos^{3} (x)) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.2.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$4 \cdot 4 {\cos (x)}^{3 + 3} + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.2.3

Some $3$ e $3$ .

$4 \cdot 4 \cos^{6} (x) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.3

Multiplique $4$ por $4$ .

$16 \cos^{6} (x) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.4

Multiplique $\cos^{3} (x)$ por $\cos (x)$ somando os expoentes.

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Etapa 4.2.4.1

Mova $\cos (x)$ .

$16 \cos^{6} (x) + 4 (\cos (x) \cos^{3} (x)) \cdot - 3 - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.4.2

Multiplique $\cos (x)$ por $\cos^{3} (x)$ .

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Etapa 4.2.4.2.1

Eleve $\cos (x)$ à potência de $1$ .

$16 \cos^{6} (x) + 4 (\cos^{1} (x) \cos^{3} (x)) \cdot - 3 - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.4.2.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$16 \cos^{6} (x) + 4 {\cos (x)}^{1 + 3} \cdot - 3 - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.4.3

Some $1$ e $3$ .

$16 \cos^{6} (x) + 4 \cos^{4} (x) \cdot - 3 - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.5

Multiplique $- 3$ por $4$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.6

Multiplique $\cos (x)$ por $\cos^{3} (x)$ somando os expoentes.

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Etapa 4.2.6.1

Mova $\cos^{3} (x)$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 3 (\cos^{3} (x) \cos (x)) \cdot 4 - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.6.2

Multiplique $\cos^{3} (x)$ por $\cos (x)$ .

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Etapa 4.2.6.2.1

Eleve $\cos (x)$ à potência de $1$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 3 (\cos^{3} (x) \cos^{1} (x)) \cdot 4 - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.6.2.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 3 {\cos (x)}^{3 + 1} \cdot 4 - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.6.3

Some $3$ e $1$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 3 \cos^{4} (x) \cdot 4 - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.7

Multiplique $4$ por $- 3$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.8

Multiplique $- 3 \cos (x) (- 3 \cos (x))$ .

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Etapa 4.2.8.1

Multiplique $- 3$ por $- 3$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos (x) \cos (x) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.8.2

Eleve $\cos (x)$ à potência de $1$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 (\cos^{1} (x) \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.8.3

Eleve $\cos (x)$ à potência de $1$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 (\cos^{1} (x) \cos^{1} (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.8.4

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 {\cos (x)}^{1 + 1} - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.8.5

Some $1$ e $1$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.9

Multiplique $4$ por $- 3$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \cos (x) \sin^{3} (x) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.10

Multiplique $- 3$ por $- 3$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \cos (x) \sin^{3} (x) + 9 \cos (x) \sin (x) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.11

Multiplique $- 3$ por $- 4$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \cos (x) \sin^{3} (x) + 9 \cos (x) \sin (x) + 12 \sin^{3} (x) \cos (x) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.12

Reescreva usando a propriedade comutativa da multiplicação.

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \cos (x) \sin^{3} (x) + 9 \cos (x) \sin (x) + 12 \sin^{3} (x) \cos (x) - 4 \cdot 4 \sin^{3} (x) \sin^{3} (x) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.13

Multiplique $\sin^{3} (x)$ por $\sin^{3} (x)$ somando os expoentes.

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Etapa 4.2.13.1

Mova $\sin^{3} (x)$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \cos (x) \sin^{3} (x) + 9 \cos (x) \sin (x) + 12 \sin^{3} (x) \cos (x) - 4 \cdot 4 (\sin^{3} (x) \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.13.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \cos (x) \sin^{3} (x) + 9 \cos (x) \sin (x) + 12 \sin^{3} (x) \cos (x) - 4 \cdot 4 {\sin (x)}^{3 + 3} - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.13.3

Some $3$ e $3$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \cos (x) \sin^{3} (x) + 9 \cos (x) \sin (x) + 12 \sin^{3} (x) \cos (x) - 4 \cdot 4 \sin^{6} (x) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.14

Multiplique $- 4$ por $4$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \cos (x) \sin^{3} (x) + 9 \cos (x) \sin (x) + 12 \sin^{3} (x) \cos (x) - 16 \sin^{6} (x) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.15

Multiplique $\sin^{3} (x)$ por $\sin (x)$ somando os expoentes.

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Etapa 4.2.15.1

Mova $\sin (x)$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \cos (x) \sin^{3} (x) + 9 \cos (x) \sin (x) + 12 \sin^{3} (x) \cos (x) - 16 \sin^{6} (x) - 4 (\sin (x) \sin^{3} (x)) \cdot - 3 + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.15.2

Multiplique $\sin (x)$ por $\sin^{3} (x)$ .

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Etapa 4.2.15.2.1

Eleve $\sin (x)$ à potência de $1$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \cos (x) \sin^{3} (x) + 9 \cos (x) \sin (x) + 12 \sin^{3} (x) \cos (x) - 16 \sin^{6} (x) - 4 (\sin^{1} (x) \sin^{3} (x)) \cdot - 3 + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.15.2.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \cos (x) \sin^{3} (x) + 9 \cos (x) \sin (x) + 12 \sin^{3} (x) \cos (x) - 16 \sin^{6} (x) - 4 {\sin (x)}^{1 + 3} \cdot - 3 + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.15.3

Some $1$ e $3$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \cos (x) \sin^{3} (x) + 9 \cos (x) \sin (x) + 12 \sin^{3} (x) \cos (x) - 16 \sin^{6} (x) - 4 \sin^{4} (x) \cdot - 3 + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.16

Multiplique $- 3$ por $- 4$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \cos (x) \sin^{3} (x) + 9 \cos (x) \sin (x) + 12 \sin^{3} (x) \cos (x) - 16 \sin^{6} (x) + 12 \sin^{4} (x) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.17

Multiplique $- 3$ por $3$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \cos (x) \sin^{3} (x) + 9 \cos (x) \sin (x) + 12 \sin^{3} (x) \cos (x) - 16 \sin^{6} (x) + 12 \sin^{4} (x) - 9 \sin (x) \cos (x) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.18

Multiplique $\sin (x)$ por $\sin^{3} (x)$ somando os expoentes.

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Etapa 4.2.18.1

Mova $\sin^{3} (x)$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \cos (x) \sin^{3} (x) + 9 \cos (x) \sin (x) + 12 \sin^{3} (x) \cos (x) - 16 \sin^{6} (x) + 12 \sin^{4} (x) - 9 \sin (x) \cos (x) + 3 (\sin^{3} (x) \sin (x)) \cdot 4 + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.18.2

Multiplique $\sin^{3} (x)$ por $\sin (x)$ .

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Etapa 4.2.18.2.1

Eleve $\sin (x)$ à potência de $1$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \cos (x) \sin^{3} (x) + 9 \cos (x) \sin (x) + 12 \sin^{3} (x) \cos (x) - 16 \sin^{6} (x) + 12 \sin^{4} (x) - 9 \sin (x) \cos (x) + 3 (\sin^{3} (x) \sin^{1} (x)) \cdot 4 + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.18.2.2

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \cos (x) \sin^{3} (x) + 9 \cos (x) \sin (x) + 12 \sin^{3} (x) \cos (x) - 16 \sin^{6} (x) + 12 \sin^{4} (x) - 9 \sin (x) \cos (x) + 3 {\sin (x)}^{3 + 1} \cdot 4 + 3 \sin (x) (- 3 \sin (x))$

Etapa 4.2.18.3

Some $3$ e $1$ .

Etapa 4.2.19

Multiplique $4$ por $3$ .

Etapa 4.2.20

Multiplique $3 \sin (x) (- 3 \sin (x))$ .

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Etapa 4.2.20.1

Multiplique $- 3$ por $3$ .

Etapa 4.2.20.2

Eleve $\sin (x)$ à potência de $1$ .

Etapa 4.2.20.3

Eleve $\sin (x)$ à potência de $1$ .

Etapa 4.2.20.4

Use a regra da multiplicação de potências $a^{m} a^{n} = a^{m + n}$ para combinar expoentes.

Etapa 4.2.20.5

Some $1$ e $1$ .

Etapa 4.3

Simplifique somando os termos.

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Etapa 4.3.1

Combine os termos opostos em $16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \cos (x) \sin^{3} (x) + 9 \cos (x) \sin (x) + 12 \sin^{3} (x) \cos (x) - 16 \sin^{6} (x) + 12 \sin^{4} (x) - 9 \sin (x) \cos (x) + 12 \sin^{4} (x) - 9 \sin^{2} (x)$ .

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Etapa 4.3.1.1

Reorganize os fatores nos termos $- 12 \cos (x) \sin^{3} (x)$ e $12 \sin^{3} (x) \cos (x)$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \sin^{3} (x) \cos (x) + 9 \cos (x) \sin (x) + 12 \sin^{3} (x) \cos (x) - 16 \sin^{6} (x) + 12 \sin^{4} (x) - 9 \sin (x) \cos (x) + 12 \sin^{4} (x) - 9 \sin^{2} (x)$

Etapa 4.3.1.2

Some $- 12 \sin^{3} (x) \cos (x)$ e $12 \sin^{3} (x) \cos (x)$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) + 9 \cos (x) \sin (x) + 0 - 16 \sin^{6} (x) + 12 \sin^{4} (x) - 9 \sin (x) \cos (x) + 12 \sin^{4} (x) - 9 \sin^{2} (x)$

Etapa 4.3.1.3

Some $16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) + 9 \cos (x) \sin (x)$ e $0$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) + 9 \cos (x) \sin (x) - 16 \sin^{6} (x) + 12 \sin^{4} (x) - 9 \sin (x) \cos (x) + 12 \sin^{4} (x) - 9 \sin^{2} (x)$

Etapa 4.3.1.4

Reorganize os fatores nos termos $9 \cos (x) \sin (x)$ e $- 9 \sin (x) \cos (x)$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) + 9 \cos (x) \sin (x) - 16 \sin^{6} (x) + 12 \sin^{4} (x) - 9 \cos (x) \sin (x) + 12 \sin^{4} (x) - 9 \sin^{2} (x)$

Etapa 4.3.1.5

Subtraia $9 \cos (x) \sin (x)$ de $9 \cos (x) \sin (x)$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) + 0 - 16 \sin^{6} (x) + 12 \sin^{4} (x) + 12 \sin^{4} (x) - 9 \sin^{2} (x)$

Etapa 4.3.1.6

Some $16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x)$ e $0$ .

$16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 16 \sin^{6} (x) + 12 \sin^{4} (x) + 12 \sin^{4} (x) - 9 \sin^{2} (x)$

Etapa 4.3.2

Subtraia $12 \cos^{4} (x)$ de $- 12 \cos^{4} (x)$ .

$16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - 16 \sin^{6} (x) + 12 \sin^{4} (x) + 12 \sin^{4} (x) - 9 \sin^{2} (x)$

Etapa 4.3.3

Some $12 \sin^{4} (x)$ e $12 \sin^{4} (x)$ .

$16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - 16 \sin^{6} (x) + 24 \sin^{4} (x) - 9 \sin^{2} (x)$