Pré-cálculo Exemplos

$\sin (4 x) \cos (6 x)$

Etapa 1

Fatore $2$ de $4 x$ .

$\sin (2 (2 x)) \cos (6 x)$

Etapa 2

Aplique a fórmula do arco duplo do seno.

$2 (2 \sin (x) \cos (x)) \cos (2 x) \cos (6 x)$

Etapa 3

Multiplique $2$ por $2$ .

$4 (\sin (x) \cos (x)) \cos (2 x) \cos (6 x)$

Etapa 4

Use a fórmula do arco duplo para transformar $\cos (2 x)$ em $2 \cos^{2} (x) - 1$ .

$4 \sin (x) \cos (x) (2 \cos^{2} (x) - 1) \cos (6 x)$

Etapa 5

Aplique a propriedade distributiva.

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

Etapa 6

Multiplique $\cos (x)$ por $\cos^{2} (x)$ somando os expoentes.

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Etapa 6.1

Mova $\cos^{2} (x)$ .

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

Etapa 6.2

Multiplique $\cos^{2} (x)$ por $\cos (x)$ .

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Etapa 6.2.1

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 6.2.2

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

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

Etapa 6.3

Some $2$ e $1$ .

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

Etapa 7

Simplifique os termos.

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Etapa 7.1

Multiplique $- 1$ por $4$ .

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

Etapa 7.2

Simplifique com fatoração.

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Etapa 7.2.1

Multiplique $2$ por $4$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) \cos (6 x)$

Etapa 7.2.2

Fatore $2$ de $6 x$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) \cos (2 (3 x))$

Etapa 7.3

Simplifique cada termo.

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Etapa 7.3.1

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

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) ({(4 \cos^{3} (x) - 3 \cos (x))}^{2} - \sin^{2} (3 x))$

Etapa 7.3.2

Reescreva ${(4 \cos^{3} (x) - 3 \cos (x))}^{2}$ como $(4 \cos^{3} (x) - 3 \cos (x)) (4 \cos^{3} (x) - 3 \cos (x))$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) ((4 \cos^{3} (x) - 3 \cos (x)) (4 \cos^{3} (x) - 3 \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.3

Expanda $(4 \cos^{3} (x) - 3 \cos (x)) (4 \cos^{3} (x) - 3 \cos (x))$ usando o método FOIL.

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Etapa 7.3.3.1

Aplique a propriedade distributiva.

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (4 \cos^{3} (x) (4 \cos^{3} (x) - 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x) - 3 \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.3.2

Aplique a propriedade distributiva.

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (4 \cos^{3} (x) (4 \cos^{3} (x)) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x) - 3 \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.3.3

Aplique a propriedade distributiva.

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (4 \cos^{3} (x) (4 \cos^{3} (x)) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.4

Simplifique e combine termos semelhantes.

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Etapa 7.3.4.1

Simplifique cada termo.

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Etapa 7.3.4.1.1

Reescreva usando a propriedade comutativa da multiplicação.

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (4 \cdot 4 \cos^{3} (x) \cos^{3} (x) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.4.1.2

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

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Etapa 7.3.4.1.2.1

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

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (4 \cdot 4 (\cos^{3} (x) \cos^{3} (x)) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.4.1.2.2

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

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (4 \cdot 4 {\cos (x)}^{3 + 3} + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.4.1.2.3

Some $3$ e $3$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (4 \cdot 4 \cos^{6} (x) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.4.1.3

Multiplique $4$ por $4$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.4.1.4

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

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Etapa 7.3.4.1.4.1

Mova $\cos (x)$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) + 4 (\cos (x) \cos^{3} (x)) \cdot - 3 - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.4.1.4.2

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

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Etapa 7.3.4.1.4.2.1

Eleve $\cos (x)$ à potência de $1$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) + 4 (\cos^{1} (x) \cos^{3} (x)) \cdot - 3 - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.4.1.4.2.2

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

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) + 4 {\cos (x)}^{1 + 3} \cdot - 3 - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.4.1.4.3

Some $1$ e $3$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) + 4 \cos^{4} (x) \cdot - 3 - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.4.1.5

Multiplique $- 3$ por $4$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 12 \cos^{4} (x) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.4.1.6

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

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Etapa 7.3.4.1.6.1

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

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 12 \cos^{4} (x) - 3 (\cos^{3} (x) \cos (x)) \cdot 4 - 3 \cos (x) (- 3 \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.4.1.6.2

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

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Etapa 7.3.4.1.6.2.1

Eleve $\cos (x)$ à potência de $1$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 12 \cos^{4} (x) - 3 (\cos^{3} (x) \cos^{1} (x)) \cdot 4 - 3 \cos (x) (- 3 \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.4.1.6.2.2

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

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 12 \cos^{4} (x) - 3 {\cos (x)}^{3 + 1} \cdot 4 - 3 \cos (x) (- 3 \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.4.1.6.3

Some $3$ e $1$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 12 \cos^{4} (x) - 3 \cos^{4} (x) \cdot 4 - 3 \cos (x) (- 3 \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.4.1.7

Multiplique $4$ por $- 3$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) - 3 \cos (x) (- 3 \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.4.1.8

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

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Etapa 7.3.4.1.8.1

Multiplique $- 3$ por $- 3$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos (x) \cos (x) - \sin^{2} (3 x))$

Etapa 7.3.4.1.8.2

Eleve $\cos (x)$ à potência de $1$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 (\cos^{1} (x) \cos (x)) - \sin^{2} (3 x))$

Etapa 7.3.4.1.8.3

Eleve $\cos (x)$ à potência de $1$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 (\cos^{1} (x) \cos^{1} (x)) - \sin^{2} (3 x))$

Etapa 7.3.4.1.8.4

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

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 {\cos (x)}^{1 + 1} - \sin^{2} (3 x))$

Etapa 7.3.4.1.8.5

Some $1$ e $1$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - \sin^{2} (3 x))$

Etapa 7.3.4.2

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

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - \sin^{2} (3 x))$

Etapa 7.3.5

Aplique a fórmula do arco triplo do seno.

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - {(- 4 \sin^{3} (x) + 3 \sin (x))}^{2})$

Etapa 7.3.6

Reescreva ${(- 4 \sin^{3} (x) + 3 \sin (x))}^{2}$ como $(- 4 \sin^{3} (x) + 3 \sin (x)) (- 4 \sin^{3} (x) + 3 \sin (x))$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - ((- 4 \sin^{3} (x) + 3 \sin (x)) (- 4 \sin^{3} (x) + 3 \sin (x))))$

Etapa 7.3.7

Expanda $(- 4 \sin^{3} (x) + 3 \sin (x)) (- 4 \sin^{3} (x) + 3 \sin (x))$ usando o método FOIL.

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Etapa 7.3.7.1

Aplique a propriedade distributiva.

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (- 4 \sin^{3} (x) (- 4 \sin^{3} (x) + 3 \sin (x)) + 3 \sin (x) (- 4 \sin^{3} (x) + 3 \sin (x))))$

Etapa 7.3.7.2

Aplique a propriedade distributiva.

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (- 4 \sin^{3} (x) (- 4 \sin^{3} (x)) - 4 \sin^{3} (x) (3 \sin (x)) + 3 \sin (x) (- 4 \sin^{3} (x) + 3 \sin (x))))$

Etapa 7.3.7.3

Aplique a propriedade distributiva.

Etapa 7.3.8

Simplifique e combine termos semelhantes.

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Etapa 7.3.8.1

Simplifique cada termo.

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Etapa 7.3.8.1.1

Reescreva usando a propriedade comutativa da multiplicação.

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (- 4 \cdot - 4 \sin^{3} (x) \sin^{3} (x) - 4 \sin^{3} (x) (3 \sin (x)) + 3 \sin (x) (- 4 \sin^{3} (x)) + 3 \sin (x) (3 \sin (x))))$

Etapa 7.3.8.1.2

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

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Etapa 7.3.8.1.2.1

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

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (- 4 \cdot - 4 (\sin^{3} (x) \sin^{3} (x)) - 4 \sin^{3} (x) (3 \sin (x)) + 3 \sin (x) (- 4 \sin^{3} (x)) + 3 \sin (x) (3 \sin (x))))$

Etapa 7.3.8.1.2.2

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

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (- 4 \cdot - 4 {\sin (x)}^{3 + 3} - 4 \sin^{3} (x) (3 \sin (x)) + 3 \sin (x) (- 4 \sin^{3} (x)) + 3 \sin (x) (3 \sin (x))))$

Etapa 7.3.8.1.2.3

Some $3$ e $3$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (- 4 \cdot - 4 \sin^{6} (x) - 4 \sin^{3} (x) (3 \sin (x)) + 3 \sin (x) (- 4 \sin^{3} (x)) + 3 \sin (x) (3 \sin (x))))$

Etapa 7.3.8.1.3

Multiplique $- 4$ por $- 4$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (16 \sin^{6} (x) - 4 \sin^{3} (x) (3 \sin (x)) + 3 \sin (x) (- 4 \sin^{3} (x)) + 3 \sin (x) (3 \sin (x))))$

Etapa 7.3.8.1.4

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

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Etapa 7.3.8.1.4.1

Mova $\sin (x)$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (16 \sin^{6} (x) - 4 (\sin (x) \sin^{3} (x)) \cdot 3 + 3 \sin (x) (- 4 \sin^{3} (x)) + 3 \sin (x) (3 \sin (x))))$

Etapa 7.3.8.1.4.2

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

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Etapa 7.3.8.1.4.2.1

Eleve $\sin (x)$ à potência de $1$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (16 \sin^{6} (x) - 4 (\sin^{1} (x) \sin^{3} (x)) \cdot 3 + 3 \sin (x) (- 4 \sin^{3} (x)) + 3 \sin (x) (3 \sin (x))))$

Etapa 7.3.8.1.4.2.2

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

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (16 \sin^{6} (x) - 4 {\sin (x)}^{1 + 3} \cdot 3 + 3 \sin (x) (- 4 \sin^{3} (x)) + 3 \sin (x) (3 \sin (x))))$

Etapa 7.3.8.1.4.3

Some $1$ e $3$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (16 \sin^{6} (x) - 4 \sin^{4} (x) \cdot 3 + 3 \sin (x) (- 4 \sin^{3} (x)) + 3 \sin (x) (3 \sin (x))))$

Etapa 7.3.8.1.5

Multiplique $3$ por $- 4$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (16 \sin^{6} (x) - 12 \sin^{4} (x) + 3 \sin (x) (- 4 \sin^{3} (x)) + 3 \sin (x) (3 \sin (x))))$

Etapa 7.3.8.1.6

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

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Etapa 7.3.8.1.6.1

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

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (16 \sin^{6} (x) - 12 \sin^{4} (x) + 3 (\sin^{3} (x) \sin (x)) \cdot - 4 + 3 \sin (x) (3 \sin (x))))$

Etapa 7.3.8.1.6.2

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

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Etapa 7.3.8.1.6.2.1

Eleve $\sin (x)$ à potência de $1$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (16 \sin^{6} (x) - 12 \sin^{4} (x) + 3 (\sin^{3} (x) \sin^{1} (x)) \cdot - 4 + 3 \sin (x) (3 \sin (x))))$

Etapa 7.3.8.1.6.2.2

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

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (16 \sin^{6} (x) - 12 \sin^{4} (x) + 3 {\sin (x)}^{3 + 1} \cdot - 4 + 3 \sin (x) (3 \sin (x))))$

Etapa 7.3.8.1.6.3

Some $3$ e $1$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (16 \sin^{6} (x) - 12 \sin^{4} (x) + 3 \sin^{4} (x) \cdot - 4 + 3 \sin (x) (3 \sin (x))))$

Etapa 7.3.8.1.7

Multiplique $- 4$ por $3$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (16 \sin^{6} (x) - 12 \sin^{4} (x) - 12 \sin^{4} (x) + 3 \sin (x) (3 \sin (x))))$

Etapa 7.3.8.1.8

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

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Etapa 7.3.8.1.8.1

Multiplique $3$ por $3$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (16 \sin^{6} (x) - 12 \sin^{4} (x) - 12 \sin^{4} (x) + 9 \sin (x) \sin (x)))$

Etapa 7.3.8.1.8.2

Eleve $\sin (x)$ à potência de $1$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (16 \sin^{6} (x) - 12 \sin^{4} (x) - 12 \sin^{4} (x) + 9 (\sin^{1} (x) \sin (x))))$

Etapa 7.3.8.1.8.3

Eleve $\sin (x)$ à potência de $1$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (16 \sin^{6} (x) - 12 \sin^{4} (x) - 12 \sin^{4} (x) + 9 (\sin^{1} (x) \sin^{1} (x))))$

Etapa 7.3.8.1.8.4

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

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (16 \sin^{6} (x) - 12 \sin^{4} (x) - 12 \sin^{4} (x) + 9 {\sin (x)}^{1 + 1}))$

Etapa 7.3.8.1.8.5

Some $1$ e $1$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (16 \sin^{6} (x) - 12 \sin^{4} (x) - 12 \sin^{4} (x) + 9 \sin^{2} (x)))$

Etapa 7.3.8.2

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

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (16 \sin^{6} (x) - 24 \sin^{4} (x) + 9 \sin^{2} (x)))$

Etapa 7.3.9

Aplique a propriedade distributiva.

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - (16 \sin^{6} (x)) - (- 24 \sin^{4} (x)) - (9 \sin^{2} (x)))$

Etapa 7.3.10

Simplifique.

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Etapa 7.3.10.1

Multiplique $16$ por $- 1$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - 16 \sin^{6} (x) - (- 24 \sin^{4} (x)) - (9 \sin^{2} (x)))$

Etapa 7.3.10.2

Multiplique $- 24$ por $- 1$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - 16 \sin^{6} (x) + 24 \sin^{4} (x) - (9 \sin^{2} (x)))$

Etapa 7.3.10.3

Multiplique $9$ por $- 1$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - 16 \sin^{6} (x) + 24 \sin^{4} (x) - 9 \sin^{2} (x))$

Etapa 8

Expanda $(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x) - 16 \sin^{6} (x) + 24 \sin^{4} (x) - 9 \sin^{2} (x))$ multiplicando cada termo na primeira expressão por cada um dos termos na segunda expressão.

$8 \sin (x) \cos^{3} (x) (16 \cos^{6} (x)) + 8 \sin (x) \cos^{3} (x) (- 24 \cos^{4} (x)) + 8 \sin (x) \cos^{3} (x) (9 \cos^{2} (x)) + 8 \sin (x) \cos^{3} (x) (- 16 \sin^{6} (x)) + 8 \sin (x) \cos^{3} (x) (24 \sin^{4} (x)) + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9

Simplifique os termos.

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Etapa 9.1

Simplifique cada termo.

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Etapa 9.1.1

Multiplique $\cos^{3} (x)$ por $\cos^{6} (x)$ somando os expoentes.

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Etapa 9.1.1.1

Mova $\cos^{6} (x)$ .

$8 \sin (x) (\cos^{6} (x) \cos^{3} (x)) \cdot 16 + 8 \sin (x) \cos^{3} (x) (- 24 \cos^{4} (x)) + 8 \sin (x) \cos^{3} (x) (9 \cos^{2} (x)) + 8 \sin (x) \cos^{3} (x) (- 16 \sin^{6} (x)) + 8 \sin (x) \cos^{3} (x) (24 \sin^{4} (x)) + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.1.2

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

$8 \sin (x) {\cos (x)}^{6 + 3} \cdot 16 + 8 \sin (x) \cos^{3} (x) (- 24 \cos^{4} (x)) + 8 \sin (x) \cos^{3} (x) (9 \cos^{2} (x)) + 8 \sin (x) \cos^{3} (x) (- 16 \sin^{6} (x)) + 8 \sin (x) \cos^{3} (x) (24 \sin^{4} (x)) + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.1.3

Some $6$ e $3$ .

$8 \sin (x) \cos^{9} (x) \cdot 16 + 8 \sin (x) \cos^{3} (x) (- 24 \cos^{4} (x)) + 8 \sin (x) \cos^{3} (x) (9 \cos^{2} (x)) + 8 \sin (x) \cos^{3} (x) (- 16 \sin^{6} (x)) + 8 \sin (x) \cos^{3} (x) (24 \sin^{4} (x)) + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.2

Multiplique $16$ por $8$ .

$128 \sin (x) \cos^{9} (x) + 8 \sin (x) \cos^{3} (x) (- 24 \cos^{4} (x)) + 8 \sin (x) \cos^{3} (x) (9 \cos^{2} (x)) + 8 \sin (x) \cos^{3} (x) (- 16 \sin^{6} (x)) + 8 \sin (x) \cos^{3} (x) (24 \sin^{4} (x)) + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.3

Multiplique $\cos^{3} (x)$ por $\cos^{4} (x)$ somando os expoentes.

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Etapa 9.1.3.1

Mova $\cos^{4} (x)$ .

$128 \sin (x) \cos^{9} (x) + 8 \sin (x) (\cos^{4} (x) \cos^{3} (x)) \cdot - 24 + 8 \sin (x) \cos^{3} (x) (9 \cos^{2} (x)) + 8 \sin (x) \cos^{3} (x) (- 16 \sin^{6} (x)) + 8 \sin (x) \cos^{3} (x) (24 \sin^{4} (x)) + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.3.2

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

$128 \sin (x) \cos^{9} (x) + 8 \sin (x) {\cos (x)}^{4 + 3} \cdot - 24 + 8 \sin (x) \cos^{3} (x) (9 \cos^{2} (x)) + 8 \sin (x) \cos^{3} (x) (- 16 \sin^{6} (x)) + 8 \sin (x) \cos^{3} (x) (24 \sin^{4} (x)) + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.3.3

Some $4$ e $3$ .

$128 \sin (x) \cos^{9} (x) + 8 \sin (x) \cos^{7} (x) \cdot - 24 + 8 \sin (x) \cos^{3} (x) (9 \cos^{2} (x)) + 8 \sin (x) \cos^{3} (x) (- 16 \sin^{6} (x)) + 8 \sin (x) \cos^{3} (x) (24 \sin^{4} (x)) + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.4

Multiplique $- 24$ por $8$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 8 \sin (x) \cos^{3} (x) (9 \cos^{2} (x)) + 8 \sin (x) \cos^{3} (x) (- 16 \sin^{6} (x)) + 8 \sin (x) \cos^{3} (x) (24 \sin^{4} (x)) + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.5

Multiplique $\cos^{3} (x)$ por $\cos^{2} (x)$ somando os expoentes.

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Etapa 9.1.5.1

Mova $\cos^{2} (x)$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 8 \sin (x) (\cos^{2} (x) \cos^{3} (x)) \cdot 9 + 8 \sin (x) \cos^{3} (x) (- 16 \sin^{6} (x)) + 8 \sin (x) \cos^{3} (x) (24 \sin^{4} (x)) + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.5.2

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

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 8 \sin (x) {\cos (x)}^{2 + 3} \cdot 9 + 8 \sin (x) \cos^{3} (x) (- 16 \sin^{6} (x)) + 8 \sin (x) \cos^{3} (x) (24 \sin^{4} (x)) + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.5.3

Some $2$ e $3$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 8 \sin (x) \cos^{5} (x) \cdot 9 + 8 \sin (x) \cos^{3} (x) (- 16 \sin^{6} (x)) + 8 \sin (x) \cos^{3} (x) (24 \sin^{4} (x)) + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.6

Multiplique $9$ por $8$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) + 8 \sin (x) \cos^{3} (x) (- 16 \sin^{6} (x)) + 8 \sin (x) \cos^{3} (x) (24 \sin^{4} (x)) + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.7

Multiplique $\sin (x)$ por $\sin^{6} (x)$ somando os expoentes.

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Etapa 9.1.7.1

Mova $\sin^{6} (x)$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) + 8 (\sin^{6} (x) \sin (x)) \cos^{3} (x) \cdot - 16 + 8 \sin (x) \cos^{3} (x) (24 \sin^{4} (x)) + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.7.2

Multiplique $\sin^{6} (x)$ por $\sin (x)$ .

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Etapa 9.1.7.2.1

Eleve $\sin (x)$ à potência de $1$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) + 8 (\sin^{6} (x) \sin^{1} (x)) \cos^{3} (x) \cdot - 16 + 8 \sin (x) \cos^{3} (x) (24 \sin^{4} (x)) + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.7.2.2

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

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) + 8 {\sin (x)}^{6 + 1} \cos^{3} (x) \cdot - 16 + 8 \sin (x) \cos^{3} (x) (24 \sin^{4} (x)) + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.7.3

Some $6$ e $1$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) + 8 \sin^{7} (x) \cos^{3} (x) \cdot - 16 + 8 \sin (x) \cos^{3} (x) (24 \sin^{4} (x)) + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.8

Multiplique $- 16$ por $8$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 8 \sin (x) \cos^{3} (x) (24 \sin^{4} (x)) + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.9

Multiplique $\sin (x)$ por $\sin^{4} (x)$ somando os expoentes.

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Etapa 9.1.9.1

Mova $\sin^{4} (x)$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 8 (\sin^{4} (x) \sin (x)) \cos^{3} (x) \cdot 24 + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.9.2

Multiplique $\sin^{4} (x)$ por $\sin (x)$ .

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Etapa 9.1.9.2.1

Eleve $\sin (x)$ à potência de $1$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 8 (\sin^{4} (x) \sin^{1} (x)) \cos^{3} (x) \cdot 24 + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.9.2.2

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

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 8 {\sin (x)}^{4 + 1} \cos^{3} (x) \cdot 24 + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.9.3

Some $4$ e $1$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 8 \sin^{5} (x) \cos^{3} (x) \cdot 24 + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.10

Multiplique $24$ por $8$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) + 8 \sin (x) \cos^{3} (x) (- 9 \sin^{2} (x)) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.11

Multiplique $\sin (x)$ por $\sin^{2} (x)$ somando os expoentes.

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Etapa 9.1.11.1

Mova $\sin^{2} (x)$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) + 8 (\sin^{2} (x) \sin (x)) \cos^{3} (x) \cdot - 9 - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.11.2

Multiplique $\sin^{2} (x)$ por $\sin (x)$ .

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Etapa 9.1.11.2.1

Eleve $\sin (x)$ à potência de $1$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) + 8 (\sin^{2} (x) \sin^{1} (x)) \cos^{3} (x) \cdot - 9 - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.11.2.2

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

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) + 8 {\sin (x)}^{2 + 1} \cos^{3} (x) \cdot - 9 - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.11.3

Some $2$ e $1$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) + 8 \sin^{3} (x) \cos^{3} (x) \cdot - 9 - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.12

Multiplique $- 9$ por $8$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 4 \sin (x) \cos (x) (16 \cos^{6} (x)) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.13

Multiplique $\cos (x)$ por $\cos^{6} (x)$ somando os expoentes.

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Etapa 9.1.13.1

Mova $\cos^{6} (x)$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 4 \sin (x) (\cos^{6} (x) \cos (x)) \cdot 16 - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.13.2

Multiplique $\cos^{6} (x)$ por $\cos (x)$ .

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Etapa 9.1.13.2.1

Eleve $\cos (x)$ à potência de $1$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 4 \sin (x) (\cos^{6} (x) \cos^{1} (x)) \cdot 16 - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.13.2.2

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

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 4 \sin (x) {\cos (x)}^{6 + 1} \cdot 16 - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.13.3

Some $6$ e $1$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 4 \sin (x) \cos^{7} (x) \cdot 16 - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.14

Multiplique $16$ por $- 4$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) - 4 \sin (x) \cos (x) (- 24 \cos^{4} (x)) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.15

Multiplique $\cos (x)$ por $\cos^{4} (x)$ somando os expoentes.

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Etapa 9.1.15.1

Mova $\cos^{4} (x)$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) - 4 \sin (x) (\cos^{4} (x) \cos (x)) \cdot - 24 - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.15.2

Multiplique $\cos^{4} (x)$ por $\cos (x)$ .

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Etapa 9.1.15.2.1

Eleve $\cos (x)$ à potência de $1$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) - 4 \sin (x) (\cos^{4} (x) \cos^{1} (x)) \cdot - 24 - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.15.2.2

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

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) - 4 \sin (x) {\cos (x)}^{4 + 1} \cdot - 24 - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.15.3

Some $4$ e $1$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) - 4 \sin (x) \cos^{5} (x) \cdot - 24 - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.16

Multiplique $- 24$ por $- 4$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 4 \sin (x) \cos (x) (9 \cos^{2} (x)) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.17

Multiplique $\cos (x)$ por $\cos^{2} (x)$ somando os expoentes.

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Etapa 9.1.17.1

Mova $\cos^{2} (x)$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 4 \sin (x) (\cos^{2} (x) \cos (x)) \cdot 9 - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.17.2

Multiplique $\cos^{2} (x)$ por $\cos (x)$ .

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Etapa 9.1.17.2.1

Eleve $\cos (x)$ à potência de $1$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 4 \sin (x) (\cos^{2} (x) \cos^{1} (x)) \cdot 9 - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.17.2.2

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

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 4 \sin (x) {\cos (x)}^{2 + 1} \cdot 9 - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.17.3

Some $2$ e $1$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 4 \sin (x) \cos^{3} (x) \cdot 9 - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.18

Multiplique $9$ por $- 4$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 36 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.19

Multiplique $\sin (x)$ por $\sin^{6} (x)$ somando os expoentes.

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Etapa 9.1.19.1

Mova $\sin^{6} (x)$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 36 \sin (x) \cos^{3} (x) - 4 (\sin^{6} (x) \sin (x)) \cos (x) \cdot - 16 - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.19.2

Multiplique $\sin^{6} (x)$ por $\sin (x)$ .

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Etapa 9.1.19.2.1

Eleve $\sin (x)$ à potência de $1$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 36 \sin (x) \cos^{3} (x) - 4 (\sin^{6} (x) \sin^{1} (x)) \cos (x) \cdot - 16 - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.19.2.2

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

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 36 \sin (x) \cos^{3} (x) - 4 {\sin (x)}^{6 + 1} \cos (x) \cdot - 16 - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.19.3

Some $6$ e $1$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 36 \sin (x) \cos^{3} (x) - 4 \sin^{7} (x) \cos (x) \cdot - 16 - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.20

Multiplique $- 16$ por $- 4$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 36 \sin (x) \cos^{3} (x) + 64 \sin^{7} (x) \cos (x) - 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.21

Multiplique $\sin (x)$ por $\sin^{4} (x)$ somando os expoentes.

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Etapa 9.1.21.1

Mova $\sin^{4} (x)$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 36 \sin (x) \cos^{3} (x) + 64 \sin^{7} (x) \cos (x) - 4 (\sin^{4} (x) \sin (x)) \cos (x) \cdot 24 - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.21.2

Multiplique $\sin^{4} (x)$ por $\sin (x)$ .

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Etapa 9.1.21.2.1

Eleve $\sin (x)$ à potência de $1$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 36 \sin (x) \cos^{3} (x) + 64 \sin^{7} (x) \cos (x) - 4 (\sin^{4} (x) \sin^{1} (x)) \cos (x) \cdot 24 - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.21.2.2

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

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 36 \sin (x) \cos^{3} (x) + 64 \sin^{7} (x) \cos (x) - 4 {\sin (x)}^{4 + 1} \cos (x) \cdot 24 - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.21.3

Some $4$ e $1$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 36 \sin (x) \cos^{3} (x) + 64 \sin^{7} (x) \cos (x) - 4 \sin^{5} (x) \cos (x) \cdot 24 - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.22

Multiplique $24$ por $- 4$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 36 \sin (x) \cos^{3} (x) + 64 \sin^{7} (x) \cos (x) - 96 \sin^{5} (x) \cos (x) - 4 \sin (x) \cos (x) (- 9 \sin^{2} (x))$

Etapa 9.1.23

Multiplique $\sin (x)$ por $\sin^{2} (x)$ somando os expoentes.

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Etapa 9.1.23.1

Mova $\sin^{2} (x)$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 36 \sin (x) \cos^{3} (x) + 64 \sin^{7} (x) \cos (x) - 96 \sin^{5} (x) \cos (x) - 4 (\sin^{2} (x) \sin (x)) \cos (x) \cdot - 9$

Etapa 9.1.23.2

Multiplique $\sin^{2} (x)$ por $\sin (x)$ .

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Etapa 9.1.23.2.1

Eleve $\sin (x)$ à potência de $1$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 36 \sin (x) \cos^{3} (x) + 64 \sin^{7} (x) \cos (x) - 96 \sin^{5} (x) \cos (x) - 4 (\sin^{2} (x) \sin^{1} (x)) \cos (x) \cdot - 9$

Etapa 9.1.23.2.2

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

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 36 \sin (x) \cos^{3} (x) + 64 \sin^{7} (x) \cos (x) - 96 \sin^{5} (x) \cos (x) - 4 {\sin (x)}^{2 + 1} \cos (x) \cdot - 9$

Etapa 9.1.23.3

Some $2$ e $1$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 36 \sin (x) \cos^{3} (x) + 64 \sin^{7} (x) \cos (x) - 96 \sin^{5} (x) \cos (x) - 4 \sin^{3} (x) \cos (x) \cdot - 9$

Etapa 9.1.24

Multiplique $- 9$ por $- 4$ .

$128 \sin (x) \cos^{9} (x) - 192 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 64 \sin (x) \cos^{7} (x) + 96 \sin (x) \cos^{5} (x) - 36 \sin (x) \cos^{3} (x) + 64 \sin^{7} (x) \cos (x) - 96 \sin^{5} (x) \cos (x) + 36 \sin^{3} (x) \cos (x)$

Etapa 9.2

Simplifique somando os termos.

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Etapa 9.2.1

Subtraia $64 \sin (x) \cos^{7} (x)$ de $- 192 \sin (x) \cos^{7} (x)$ .

$128 \sin (x) \cos^{9} (x) - 256 \sin (x) \cos^{7} (x) + 72 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) + 96 \sin (x) \cos^{5} (x) - 36 \sin (x) \cos^{3} (x) + 64 \sin^{7} (x) \cos (x) - 96 \sin^{5} (x) \cos (x) + 36 \sin^{3} (x) \cos (x)$

Etapa 9.2.2

Some $72 \sin (x) \cos^{5} (x)$ e $96 \sin (x) \cos^{5} (x)$ .

$128 \sin (x) \cos^{9} (x) - 256 \sin (x) \cos^{7} (x) + 168 \sin (x) \cos^{5} (x) - 128 \sin^{7} (x) \cos^{3} (x) + 192 \sin^{5} (x) \cos^{3} (x) - 72 \sin^{3} (x) \cos^{3} (x) - 36 \sin (x) \cos^{3} (x) + 64 \sin^{7} (x) \cos (x) - 96 \sin^{5} (x) \cos (x) + 36 \sin^{3} (x) \cos (x)$