Trigonometria Exemplos

$\sin (8 x) - \sin (6 x)$

Etapa 1

Simplifique cada termo.

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Etapa 1.1

Fatore $2$ de $8 x$ .

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

Etapa 1.2

Fatore $2$ de $4 x$ .

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

Etapa 1.3

Aplique a fórmula do arco duplo do seno.

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

Etapa 1.4

Multiplique $2$ por $2$ .

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

Etapa 1.5

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

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

Etapa 1.6

Aplique a propriedade distributiva.

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

Etapa 1.7

Multiplique $4$ por $1$ .

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

Etapa 1.8

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

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Etapa 1.8.1

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

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

Etapa 1.8.2

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

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Etapa 1.8.2.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 1.8.2.2

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

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

Etapa 1.8.3

Some $2$ e $1$ .

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

Etapa 1.9

Multiplique $- 2$ por $4$ .

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

Etapa 1.10

Aplique a propriedade distributiva.

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

Etapa 1.11

Multiplique $4$ por $2$ .

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

Etapa 1.12

Multiplique $- 8$ por $2$ .

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

Etapa 1.13

Remova os parênteses.

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

Etapa 1.14

Fatore $2$ de $4 x$ .

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

Etapa 1.15

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Etapa 1.15.1

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

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

Etapa 1.15.2

Reescreva ${(1 - 2 \sin^{2} (x))}^{2}$ como $(1 - 2 \sin^{2} (x)) (1 - 2 \sin^{2} (x))$ .

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

Etapa 1.15.3

Expanda $(1 - 2 \sin^{2} (x)) (1 - 2 \sin^{2} (x))$ usando o método FOIL.

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Etapa 1.15.3.1

Aplique a propriedade distributiva.

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

Etapa 1.15.3.2

Aplique a propriedade distributiva.

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

Etapa 1.15.3.3

Aplique a propriedade distributiva.

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

Etapa 1.15.4

Simplifique e combine termos semelhantes.

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Etapa 1.15.4.1

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Etapa 1.15.4.1.1

Multiplique $1$ por $1$ .

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

Etapa 1.15.4.1.2

Multiplique $- 2 \sin^{2} (x)$ por $1$ .

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

Etapa 1.15.4.1.3

Multiplique $- 2$ por $1$ .

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

Etapa 1.15.4.1.4

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 1.15.4.1.5

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

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Etapa 1.15.4.1.5.1

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

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

Etapa 1.15.4.1.5.2

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

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

Etapa 1.15.4.1.5.3

Some $2$ e $2$ .

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

Etapa 1.15.4.1.6

Multiplique $- 2$ por $- 2$ .

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

Etapa 1.15.4.2

Subtraia $2 \sin^{2} (x)$ de $- 2 \sin^{2} (x)$ .

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

Etapa 1.15.5

Aplique a fórmula do arco duplo do seno.

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

Etapa 1.15.6

Use a regra da multiplicação de potências ${(a b)}^{n} = a^{n} b^{n}$ para distribuir o expoente.

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Etapa 1.15.6.1

Aplique a regra do produto a $2 \sin (x) \cos (x)$ .

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

Etapa 1.15.6.2

Aplique a regra do produto a $2 \sin (x)$ .

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

Etapa 1.15.7

Eleve $2$ à potência de $2$ .

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

Etapa 1.15.8

Multiplique $4$ por $- 1$ .

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

Etapa 1.16

Expanda $(8 \sin (x) \cos (x) - 16 \sin^{3} (x) \cos (x)) (1 - 4 \sin^{2} (x) + 4 \sin^{4} (x) - 4 \sin^{2} (x) \cos^{2} (x))$ multiplicando cada termo na primeira expressão por cada um dos termos na segunda expressão.

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

Etapa 1.17

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Etapa 1.17.1

Multiplique $8$ por $1$ .

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

Etapa 1.17.2

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

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Etapa 1.17.2.1

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

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

Etapa 1.17.2.2

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

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Etapa 1.17.2.2.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 1.17.2.2.2

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

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

Etapa 1.17.2.3

Some $2$ e $1$ .

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

Etapa 1.17.3

Multiplique $- 4$ por $8$ .

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 8 \sin (x) \cos (x) (4 \sin^{4} (x)) + 8 \sin (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - 16 \sin^{3} (x) \cos (x) \cdot 1 - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x)) - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.4

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

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Etapa 1.17.4.1

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

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 8 (\sin^{4} (x) \sin (x)) \cos (x) \cdot 4 + 8 \sin (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - 16 \sin^{3} (x) \cos (x) \cdot 1 - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x)) - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.4.2

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

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Etapa 1.17.4.2.1

Eleve $\sin (x)$ à potência de $1$ .

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 8 (\sin^{4} (x) \sin^{1} (x)) \cos (x) \cdot 4 + 8 \sin (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - 16 \sin^{3} (x) \cos (x) \cdot 1 - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x)) - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.4.2.2

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

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 8 {\sin (x)}^{4 + 1} \cos (x) \cdot 4 + 8 \sin (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - 16 \sin^{3} (x) \cos (x) \cdot 1 - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x)) - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.4.3

Some $4$ e $1$ .

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 8 \sin^{5} (x) \cos (x) \cdot 4 + 8 \sin (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - 16 \sin^{3} (x) \cos (x) \cdot 1 - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x)) - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.5

Multiplique $4$ por $8$ .

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) + 8 \sin (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - 16 \sin^{3} (x) \cos (x) \cdot 1 - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x)) - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.6

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

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Etapa 1.17.6.1

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

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) + 8 (\sin^{2} (x) \sin (x)) \cos (x) (- 4 \cos^{2} (x)) - 16 \sin^{3} (x) \cos (x) \cdot 1 - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x)) - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.6.2

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

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Etapa 1.17.6.2.1

Eleve $\sin (x)$ à potência de $1$ .

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) + 8 (\sin^{2} (x) \sin^{1} (x)) \cos (x) (- 4 \cos^{2} (x)) - 16 \sin^{3} (x) \cos (x) \cdot 1 - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x)) - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.6.2.2

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

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) + 8 {\sin (x)}^{2 + 1} \cos (x) (- 4 \cos^{2} (x)) - 16 \sin^{3} (x) \cos (x) \cdot 1 - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x)) - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.6.3

Some $2$ e $1$ .

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) + 8 \sin^{3} (x) \cos (x) (- 4 \cos^{2} (x)) - 16 \sin^{3} (x) \cos (x) \cdot 1 - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x)) - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.7

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

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Etapa 1.17.7.1

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

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) + 8 \sin^{3} (x) (\cos^{2} (x) \cos (x)) \cdot - 4 - 16 \sin^{3} (x) \cos (x) \cdot 1 - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x)) - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.7.2

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

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Etapa 1.17.7.2.1

Eleve $\cos (x)$ à potência de $1$ .

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) + 8 \sin^{3} (x) (\cos^{2} (x) \cos^{1} (x)) \cdot - 4 - 16 \sin^{3} (x) \cos (x) \cdot 1 - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x)) - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.7.2.2

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

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) + 8 \sin^{3} (x) {\cos (x)}^{2 + 1} \cdot - 4 - 16 \sin^{3} (x) \cos (x) \cdot 1 - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x)) - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.7.3

Some $2$ e $1$ .

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) + 8 \sin^{3} (x) \cos^{3} (x) \cdot - 4 - 16 \sin^{3} (x) \cos (x) \cdot 1 - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x)) - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.8

Multiplique $- 4$ por $8$ .

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 16 \sin^{3} (x) \cos (x) \cdot 1 - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x)) - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.9

Multiplique $- 16$ por $1$ .

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 16 \sin^{3} (x) \cos (x) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x)) - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.10

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

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Etapa 1.17.10.1

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

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 16 \sin^{3} (x) \cos (x) - 16 (\sin^{2} (x) \sin^{3} (x)) \cos (x) \cdot - 4 - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.10.2

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

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 16 \sin^{3} (x) \cos (x) - 16 {\sin (x)}^{2 + 3} \cos (x) \cdot - 4 - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.10.3

Some $2$ e $3$ .

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 16 \sin^{3} (x) \cos (x) - 16 \sin^{5} (x) \cos (x) \cdot - 4 - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.11

Multiplique $- 4$ por $- 16$ .

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 16 \sin^{3} (x) \cos (x) + 64 \sin^{5} (x) \cos (x) - 16 \sin^{3} (x) \cos (x) (4 \sin^{4} (x)) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.12

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

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Etapa 1.17.12.1

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

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 16 \sin^{3} (x) \cos (x) + 64 \sin^{5} (x) \cos (x) - 16 (\sin^{4} (x) \sin^{3} (x)) \cos (x) \cdot 4 - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.12.2

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

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 16 \sin^{3} (x) \cos (x) + 64 \sin^{5} (x) \cos (x) - 16 {\sin (x)}^{4 + 3} \cos (x) \cdot 4 - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.12.3

Some $4$ e $3$ .

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 16 \sin^{3} (x) \cos (x) + 64 \sin^{5} (x) \cos (x) - 16 \sin^{7} (x) \cos (x) \cdot 4 - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.13

Multiplique $4$ por $- 16$ .

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 16 \sin^{3} (x) \cos (x) + 64 \sin^{5} (x) \cos (x) - 64 \sin^{7} (x) \cos (x) - 16 \sin^{3} (x) \cos (x) (- 4 \sin^{2} (x) \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.14

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

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Etapa 1.17.14.1

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

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 16 \sin^{3} (x) \cos (x) + 64 \sin^{5} (x) \cos (x) - 64 \sin^{7} (x) \cos (x) - 16 (\sin^{2} (x) \sin^{3} (x)) \cos (x) (- 4 \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.14.2

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

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 16 \sin^{3} (x) \cos (x) + 64 \sin^{5} (x) \cos (x) - 64 \sin^{7} (x) \cos (x) - 16 {\sin (x)}^{2 + 3} \cos (x) (- 4 \cos^{2} (x)) - \sin (6 x)$

Etapa 1.17.14.3

Some $2$ e $3$ .

Etapa 1.17.15

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

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Etapa 1.17.15.1

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

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 16 \sin^{3} (x) \cos (x) + 64 \sin^{5} (x) \cos (x) - 64 \sin^{7} (x) \cos (x) - 16 \sin^{5} (x) (\cos^{2} (x) \cos (x)) \cdot - 4 - \sin (6 x)$

Etapa 1.17.15.2

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

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Etapa 1.17.15.2.1

Eleve $\cos (x)$ à potência de $1$ .

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 16 \sin^{3} (x) \cos (x) + 64 \sin^{5} (x) \cos (x) - 64 \sin^{7} (x) \cos (x) - 16 \sin^{5} (x) (\cos^{2} (x) \cos^{1} (x)) \cdot - 4 - \sin (6 x)$

Etapa 1.17.15.2.2

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

Etapa 1.17.15.3

Some $2$ e $1$ .

Etapa 1.17.16

Multiplique $- 4$ por $- 16$ .

$8 \sin (x) \cos (x) - 32 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 16 \sin^{3} (x) \cos (x) + 64 \sin^{5} (x) \cos (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - \sin (6 x)$

Etapa 1.18

Subtraia $16 \sin^{3} (x) \cos (x)$ de $- 32 \sin^{3} (x) \cos (x)$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 32 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) + 64 \sin^{5} (x) \cos (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - \sin (6 x)$

Etapa 1.19

Some $32 \sin^{5} (x) \cos (x)$ e $64 \sin^{5} (x) \cos (x)$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - \sin (6 x)$

Etapa 1.20

Fatore $2$ de $6 x$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - \sin (2 (3 x))$

Etapa 1.21

Aplique a fórmula do arco triplo do seno.

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (2 (- 4 \sin^{3} (x) + 3 \sin (x)) \cos (3 x))$

Etapa 1.22

Aplique a propriedade distributiva.

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - ((2 (- 4 \sin^{3} (x)) + 2 (3 \sin (x))) \cos (3 x))$

Etapa 1.23

Multiplique $- 4$ por $2$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - ((- 8 \sin^{3} (x) + 2 (3 \sin (x))) \cos (3 x))$

Etapa 1.24

Multiplique $3$ por $2$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - ((- 8 \sin^{3} (x) + 6 \sin (x)) \cos (3 x))$

Etapa 1.25

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

Etapa 1.26

Expanda $(- 8 \sin^{3} (x) + 6 \sin (x)) (4 \cos^{3} (x) - 3 \cos (x))$ usando o método FOIL.

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Etapa 1.26.1

Aplique a propriedade distributiva.

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (- 8 \sin^{3} (x) (4 \cos^{3} (x) - 3 \cos (x)) + 6 \sin (x) (4 \cos^{3} (x) - 3 \cos (x)))$

Etapa 1.26.2

Aplique a propriedade distributiva.

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (- 8 \sin^{3} (x) (4 \cos^{3} (x)) - 8 \sin^{3} (x) (- 3 \cos (x)) + 6 \sin (x) (4 \cos^{3} (x) - 3 \cos (x)))$

Etapa 1.26.3

Aplique a propriedade distributiva.

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (- 8 \sin^{3} (x) (4 \cos^{3} (x)) - 8 \sin^{3} (x) (- 3 \cos (x)) + 6 \sin (x) (4 \cos^{3} (x)) + 6 \sin (x) (- 3 \cos (x)))$

Etapa 1.27

Simplifique cada termo.

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Etapa 1.27.1

Reescreva usando a propriedade comutativa da multiplicação.

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (- 8 \cdot 4 \sin^{3} (x) \cos^{3} (x) - 8 \sin^{3} (x) (- 3 \cos (x)) + 6 \sin (x) (4 \cos^{3} (x)) + 6 \sin (x) (- 3 \cos (x)))$

Etapa 1.27.2

Multiplique $- 8$ por $4$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (- 32 \sin^{3} (x) \cos^{3} (x) - 8 \sin^{3} (x) (- 3 \cos (x)) + 6 \sin (x) (4 \cos^{3} (x)) + 6 \sin (x) (- 3 \cos (x)))$

Etapa 1.27.3

Multiplique $- 3$ por $- 8$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (- 32 \sin^{3} (x) \cos^{3} (x) + 24 \sin^{3} (x) \cos (x) + 6 \sin (x) (4 \cos^{3} (x)) + 6 \sin (x) (- 3 \cos (x)))$

Etapa 1.27.4

Multiplique $4$ por $6$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (- 32 \sin^{3} (x) \cos^{3} (x) + 24 \sin^{3} (x) \cos (x) + 24 \sin (x) \cos^{3} (x) + 6 \sin (x) (- 3 \cos (x)))$

Etapa 1.27.5

Multiplique $- 3$ por $6$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (- 32 \sin^{3} (x) \cos^{3} (x) + 24 \sin^{3} (x) \cos (x) + 24 \sin (x) \cos^{3} (x) - 18 \sin (x) \cos (x))$

Etapa 1.28

Fatore $2 \sin (x) \cos (x)$ de $- 32 \sin^{3} (x) \cos^{3} (x) + 24 \sin^{3} (x) \cos (x) + 24 \sin (x) \cos^{3} (x) - 18 \sin (x) \cos (x)$ .

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Etapa 1.28.1

Fatore $2 \sin (x) \cos (x)$ de $- 32 \sin^{3} (x) \cos^{3} (x)$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x)) + 24 \sin^{3} (x) \cos (x) + 24 \sin (x) \cos^{3} (x) - 18 \sin (x) \cos (x))$

Etapa 1.28.2

Fatore $2 \sin (x) \cos (x)$ de $24 \sin^{3} (x) \cos (x)$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x)) + 2 \sin (x) \cos (x) (12 \sin^{2} (x)) + 24 \sin (x) \cos^{3} (x) - 18 \sin (x) \cos (x))$

Etapa 1.28.3

Fatore $2 \sin (x) \cos (x)$ de $24 \sin (x) \cos^{3} (x)$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x)) + 2 \sin (x) \cos (x) (12 \sin^{2} (x)) + 2 \sin (x) \cos (x) (12 \cos^{2} (x)) - 18 \sin (x) \cos (x))$

Etapa 1.28.4

Fatore $2 \sin (x) \cos (x)$ de $- 18 \sin (x) \cos (x)$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x)) + 2 \sin (x) \cos (x) (12 \sin^{2} (x)) + 2 \sin (x) \cos (x) (12 \cos^{2} (x)) + 2 \sin (x) \cos (x) (- 9))$

Etapa 1.28.5

Fatore $2 \sin (x) \cos (x)$ de $2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x)) + 2 \sin (x) \cos (x) (12 \sin^{2} (x))$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x) + 12 \sin^{2} (x)) + 2 \sin (x) \cos (x) (12 \cos^{2} (x)) + 2 \sin (x) \cos (x) (- 9))$

Etapa 1.28.6

Fatore $2 \sin (x) \cos (x)$ de $2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x) + 12 \sin^{2} (x)) + 2 \sin (x) \cos (x) (12 \cos^{2} (x))$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x) + 12 \sin^{2} (x) + 12 \cos^{2} (x)) + 2 \sin (x) \cos (x) (- 9))$

Etapa 1.28.7

Fatore $2 \sin (x) \cos (x)$ de $2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x) + 12 \sin^{2} (x) + 12 \cos^{2} (x)) + 2 \sin (x) \cos (x) (- 9)$ .

Etapa 1.29

Fatore $12$ de $12 \sin^{2} (x)$ .

Etapa 1.30

Fatore $12$ de $12 \cos^{2} (x)$ .

Etapa 1.31

Fatore $12$ de $12 (\sin^{2} (x)) + 12 (\cos^{2} (x))$ .

Etapa 1.32

Aplique a identidade trigonométrica fundamental.

Etapa 1.33

Multiplique $12$ por $1$ .

Etapa 1.34

Subtraia $9$ de $12$ .

Etapa 1.35

Aplique a propriedade distributiva.

Etapa 1.36

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

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Etapa 1.36.1

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

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (2 (\sin^{2} (x) \sin (x)) \cos (x) (- 16 \cos^{2} (x)) + 2 \sin (x) \cos (x) \cdot 3)$

Etapa 1.36.2

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

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Etapa 1.36.2.1

Eleve $\sin (x)$ à potência de $1$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (2 (\sin^{2} (x) \sin^{1} (x)) \cos (x) (- 16 \cos^{2} (x)) + 2 \sin (x) \cos (x) \cdot 3)$

Etapa 1.36.2.2

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

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (2 {\sin (x)}^{2 + 1} \cos (x) (- 16 \cos^{2} (x)) + 2 \sin (x) \cos (x) \cdot 3)$

Etapa 1.36.3

Some $2$ e $1$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (2 \sin^{3} (x) \cos (x) (- 16 \cos^{2} (x)) + 2 \sin (x) \cos (x) \cdot 3)$

Etapa 1.37

Multiplique $3$ por $2$ .

Etapa 1.38

Simplifique cada termo.

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Etapa 1.38.1

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

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Etapa 1.38.1.1

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

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (2 \sin^{3} (x) (\cos^{2} (x) \cos (x)) \cdot - 16 + 6 \sin (x) \cos (x))$

Etapa 1.38.1.2

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

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Etapa 1.38.1.2.1

Eleve $\cos (x)$ à potência de $1$ .

Etapa 1.38.1.2.2

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

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (2 \sin^{3} (x) {\cos (x)}^{2 + 1} \cdot - 16 + 6 \sin (x) \cos (x))$

Etapa 1.38.1.3

Some $2$ e $1$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - (2 \sin^{3} (x) \cos^{3} (x) \cdot - 16 + 6 \sin (x) \cos (x))$

Etapa 1.38.2

Multiplique $- 16$ por $2$ .

Etapa 1.39

Aplique a propriedade distributiva.

Etapa 1.40

Multiplique $- 32$ por $- 1$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) + 32 (\sin^{3} (x) \cos^{3} (x)) - (6 \sin (x) \cos (x))$

Etapa 1.41

Multiplique $6$ por $- 1$ .

Etapa 1.42

Remova os parênteses.

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) + 32 \sin^{3} (x) \cos^{3} (x) - 6 \sin (x) \cos (x)$

Etapa 2

Simplifique somando os termos.

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Etapa 2.1

Combine os termos opostos em $8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) + 32 \sin^{3} (x) \cos^{3} (x) - 6 \sin (x) \cos (x)$ .

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Etapa 2.1.1

Some $- 32 \sin^{3} (x) \cos^{3} (x)$ e $32 \sin^{3} (x) \cos^{3} (x)$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) + 0 - 6 \sin (x) \cos (x)$

Etapa 2.1.2

Some $8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x)$ e $0$ .

$8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) - 6 \sin (x) \cos (x)$

Etapa 2.2

Subtraia $6 \sin (x) \cos (x)$ de $8 \sin (x) \cos (x)$ .

$2 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x)$