Pré-cálculo Exemplos

$\sin (4 x) + \sin (8 x)$

Etapa 1

Simplifique os termos.

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Etapa 1.1

Simplifique cada termo.

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Etapa 1.1.1

Fatore $2$ de $4 x$ .

$\sin (2 (2 x)) + \sin (8 x)$

Etapa 1.1.2

Aplique a fórmula do arco duplo do seno.

$2 (2 \sin (x) \cos (x)) \cos (2 x) + \sin (8 x)$

Etapa 1.1.3

Multiplique $2$ por $2$ .

$4 (\sin (x) \cos (x)) \cos (2 x) + \sin (8 x)$

Etapa 1.1.4

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

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

Etapa 1.1.5

Aplique a propriedade distributiva.

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

Etapa 1.1.6

Multiplique $4$ por $1$ .

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

Etapa 1.1.7

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

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Etapa 1.1.7.1

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

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

Etapa 1.1.7.2

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

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Etapa 1.1.7.2.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 1.1.7.2.2

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

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

Etapa 1.1.7.3

Some $2$ e $1$ .

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

Etapa 1.1.8

Multiplique $- 2$ por $4$ .

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

Etapa 1.1.9

Fatore $2$ de $8 x$ .

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

Etapa 1.1.10

Fatore $2$ de $4 x$ .

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

Etapa 1.1.11

Aplique a fórmula do arco duplo do seno.

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

Etapa 1.1.12

Multiplique $2$ por $2$ .

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

Etapa 1.1.13

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

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

Etapa 1.1.14

Aplique a propriedade distributiva.

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

Etapa 1.1.15

Multiplique $4$ por $1$ .

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

Etapa 1.1.16

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

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Etapa 1.1.16.1

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

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

Etapa 1.1.16.2

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

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Etapa 1.1.16.2.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 1.1.16.2.2

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

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

Etapa 1.1.16.3

Some $2$ e $1$ .

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

Etapa 1.1.17

Multiplique $- 2$ por $4$ .

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

Etapa 1.1.18

Aplique a propriedade distributiva.

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

Etapa 1.1.19

Multiplique $4$ por $2$ .

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

Etapa 1.1.20

Multiplique $- 8$ por $2$ .

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

Etapa 1.1.21

Remova os parênteses.

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

Etapa 1.1.22

Fatore $2$ de $4 x$ .

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

Etapa 1.1.23

Simplifique cada termo.

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Etapa 1.1.23.1

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

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

Etapa 1.1.23.2

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

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

Etapa 1.1.23.3

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

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Etapa 1.1.23.3.1

Aplique a propriedade distributiva.

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

Etapa 1.1.23.3.2

Aplique a propriedade distributiva.

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

Etapa 1.1.23.3.3

Aplique a propriedade distributiva.

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

Etapa 1.1.23.4

Simplifique e combine termos semelhantes.

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Etapa 1.1.23.4.1

Simplifique cada termo.

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Etapa 1.1.23.4.1.1

Multiplique $1$ por $1$ .

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

Etapa 1.1.23.4.1.2

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

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

Etapa 1.1.23.4.1.3

Multiplique $- 2$ por $1$ .

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

Etapa 1.1.23.4.1.4

Reescreva usando a propriedade comutativa da multiplicação.

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

Etapa 1.1.23.4.1.5

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

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Etapa 1.1.23.4.1.5.1

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

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

Etapa 1.1.23.4.1.5.2

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

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

Etapa 1.1.23.4.1.5.3

Some $2$ e $2$ .

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

Etapa 1.1.23.4.1.6

Multiplique $- 2$ por $- 2$ .

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

Etapa 1.1.23.4.2

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

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

Etapa 1.1.23.5

Aplique a fórmula do arco duplo do seno.

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

Etapa 1.1.23.6

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

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Etapa 1.1.23.6.1

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

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

Etapa 1.1.23.6.2

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

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

Etapa 1.1.23.7

Eleve $2$ à potência de $2$ .

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

Etapa 1.1.23.8

Multiplique $4$ por $- 1$ .

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

Etapa 1.1.24

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.

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

Etapa 1.1.25

Simplifique cada termo.

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Etapa 1.1.25.1

Multiplique $8$ por $1$ .

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

Etapa 1.1.25.2

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

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Etapa 1.1.25.2.1

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

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

Etapa 1.1.25.2.2

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

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Etapa 1.1.25.2.2.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 1.1.25.2.2.2

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

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

Etapa 1.1.25.2.3

Some $2$ e $1$ .

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

Etapa 1.1.25.3

Multiplique $- 4$ por $8$ .

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

Etapa 1.1.25.4

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

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Etapa 1.1.25.4.1

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

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

Etapa 1.1.25.4.2

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

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Etapa 1.1.25.4.2.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 1.1.25.4.2.2

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

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

Etapa 1.1.25.4.3

Some $4$ e $1$ .

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

Etapa 1.1.25.5

Multiplique $4$ por $8$ .

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

Etapa 1.1.25.6

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

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Etapa 1.1.25.6.1

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

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

Etapa 1.1.25.6.2

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

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Etapa 1.1.25.6.2.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 1.1.25.6.2.2

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

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

Etapa 1.1.25.6.3

Some $2$ e $1$ .

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

Etapa 1.1.25.7

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

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Etapa 1.1.25.7.1

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

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

Etapa 1.1.25.7.2

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

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Etapa 1.1.25.7.2.1

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 1.1.25.7.2.2

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

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

Etapa 1.1.25.7.3

Some $2$ e $1$ .

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

Etapa 1.1.25.8

Multiplique $- 4$ por $8$ .

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

Etapa 1.1.25.9

Multiplique $- 16$ por $1$ .

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

Etapa 1.1.25.10

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

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Etapa 1.1.25.10.1

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

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

Etapa 1.1.25.10.2

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

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

Etapa 1.1.25.10.3

Some $2$ e $3$ .

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

Etapa 1.1.25.11

Multiplique $- 4$ por $- 16$ .

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

Etapa 1.1.25.12

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

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Etapa 1.1.25.12.1

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

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

Etapa 1.1.25.12.2

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

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

Etapa 1.1.25.12.3

Some $4$ e $3$ .

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

Etapa 1.1.25.13

Multiplique $4$ por $- 16$ .

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

Etapa 1.1.25.14

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

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Etapa 1.1.25.14.1

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

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

Etapa 1.1.25.14.2

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

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

Etapa 1.1.25.14.3

Some $2$ e $3$ .

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

Etapa 1.1.25.15

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

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Etapa 1.1.25.15.1

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

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

Etapa 1.1.25.15.2

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

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Etapa 1.1.25.15.2.1

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 1.1.25.15.2.2

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

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

Etapa 1.1.25.15.3

Some $2$ e $1$ .

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

Etapa 1.1.25.16

Multiplique $- 4$ por $- 16$ .

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

Etapa 1.1.26

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

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

Etapa 1.1.27

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

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

Etapa 1.2

Simplifique com fatoração.

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Etapa 1.2.1

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

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Etapa 1.2.1.1

Fatore $4 \sin (x) \cos (x)$ de $4 \sin (x) \cos (x)$ .

$4 \sin (x) \cos (x) (1) - 8 \sin^{3} (x) \cos (x) + 8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x)$

Etapa 1.2.1.2

Fatore $4 \sin (x) \cos (x)$ de $- 8 \sin^{3} (x) \cos (x)$ .

$4 \sin (x) \cos (x) (1) + 4 \sin (x) \cos (x) (- 2 \sin^{2} (x)) + 8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x)$

Etapa 1.2.1.3

Fatore $4 \sin (x) \cos (x)$ de $8 \sin (x) \cos (x)$ .

$4 \sin (x) \cos (x) (1) + 4 \sin (x) \cos (x) (- 2 \sin^{2} (x)) + 4 \sin (x) \cos (x) (2) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x)$

Etapa 1.2.1.4

Fatore $4 \sin (x) \cos (x)$ de $- 48 \sin^{3} (x) \cos (x)$ .

$4 \sin (x) \cos (x) (1) + 4 \sin (x) \cos (x) (- 2 \sin^{2} (x)) + 4 \sin (x) \cos (x) (2) + 4 \sin (x) \cos (x) (- 12 \sin^{2} (x)) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x)$

Etapa 1.2.1.5

Fatore $4 \sin (x) \cos (x)$ de $96 \sin^{5} (x) \cos (x)$ .

$4 \sin (x) \cos (x) (1) + 4 \sin (x) \cos (x) (- 2 \sin^{2} (x)) + 4 \sin (x) \cos (x) (2) + 4 \sin (x) \cos (x) (- 12 \sin^{2} (x)) + 4 \sin (x) \cos (x) (24 \sin^{4} (x)) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x)$

Etapa 1.2.1.6

Fatore $4 \sin (x) \cos (x)$ de $- 32 \sin^{3} (x) \cos^{3} (x)$ .

$4 \sin (x) \cos (x) (1) + 4 \sin (x) \cos (x) (- 2 \sin^{2} (x)) + 4 \sin (x) \cos (x) (2) + 4 \sin (x) \cos (x) (- 12 \sin^{2} (x)) + 4 \sin (x) \cos (x) (24 \sin^{4} (x)) + 4 \sin (x) \cos (x) (- 8 \sin^{2} (x) \cos^{2} (x)) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x)$

Etapa 1.2.1.7

Fatore $4 \sin (x) \cos (x)$ de $- 64 \sin^{7} (x) \cos (x)$ .

$4 \sin (x) \cos (x) (1) + 4 \sin (x) \cos (x) (- 2 \sin^{2} (x)) + 4 \sin (x) \cos (x) (2) + 4 \sin (x) \cos (x) (- 12 \sin^{2} (x)) + 4 \sin (x) \cos (x) (24 \sin^{4} (x)) + 4 \sin (x) \cos (x) (- 8 \sin^{2} (x) \cos^{2} (x)) + 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) + 64 \sin^{5} (x) \cos^{3} (x)$

Etapa 1.2.1.8

Fatore $4 \sin (x) \cos (x)$ de $64 \sin^{5} (x) \cos^{3} (x)$ .

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

Etapa 1.2.1.9

Fatore $4 \sin (x) \cos (x)$ de $4 \sin (x) \cos (x) (1) + 4 \sin (x) \cos (x) (- 2 \sin^{2} (x))$ .

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

Etapa 1.2.1.10

Fatore $4 \sin (x) \cos (x)$ de $4 \sin (x) \cos (x) (1 - 2 \sin^{2} (x)) + 4 \sin (x) \cos (x) (2)$ .

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

Etapa 1.2.1.11

Fatore $4 \sin (x) \cos (x)$ de $4 \sin (x) \cos (x) (1 - 2 \sin^{2} (x) + 2) + 4 \sin (x) \cos (x) (- 12 \sin^{2} (x))$ .

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

Etapa 1.2.1.12

Fatore $4 \sin (x) \cos (x)$ de $4 \sin (x) \cos (x) (1 - 2 \sin^{2} (x) + 2 - 12 \sin^{2} (x)) + 4 \sin (x) \cos (x) (24 \sin^{4} (x))$ .

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

Etapa 1.2.1.13

Fatore $4 \sin (x) \cos (x)$ de $4 \sin (x) \cos (x) (1 - 2 \sin^{2} (x) + 2 - 12 \sin^{2} (x) + 24 \sin^{4} (x)) + 4 \sin (x) \cos (x) (- 8 \sin^{2} (x) \cos^{2} (x))$ .

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

Etapa 1.2.1.14

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

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

Etapa 1.2.1.15

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

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

Etapa 1.2.2

Reordene $- 2 \sin^{2} (x)$ e $2$ .

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

Etapa 1.2.3

Fatore $2$ de $2$ .

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

Etapa 1.2.4

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

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

Etapa 1.2.5

Fatore $2$ de $2 (1) + 2 (- \sin^{2} (x))$ .

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

Etapa 2

Aplique a identidade trigonométrica fundamental.

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

Etapa 3

Aplique a propriedade distributiva.

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

Etapa 4

Simplifique.

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Etapa 4.1

Multiplique $4$ por $1$ .

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

Etapa 4.2

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

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Etapa 4.2.1

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

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

Etapa 4.2.2

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

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Etapa 4.2.2.1

Eleve $\cos (x)$ à potência de $1$ .

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

Etapa 4.2.2.2

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

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

Etapa 4.2.3

Some $2$ e $1$ .

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

Etapa 4.3

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

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Etapa 4.3.1

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

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

Etapa 4.3.2

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

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Etapa 4.3.2.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 4.3.2.2

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

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

Etapa 4.3.3

Some $2$ e $1$ .

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

Etapa 4.4

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

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Etapa 4.4.1

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

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

Etapa 4.4.2

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

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Etapa 4.4.2.1

Eleve $\sin (x)$ à potência de $1$ .

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

Etapa 4.4.2.2

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

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

Etapa 4.4.3

Some $4$ e $1$ .

$4 \sin (x) \cos (x) + 4 \sin (x) \cos^{3} (x) \cdot 2 + 4 \sin^{3} (x) \cos (x) \cdot - 12 + 4 \sin^{5} (x) \cos (x) \cdot 24 + 4 \sin (x) \cos (x) (- 8 \sin^{2} (x) \cos^{2} (x)) + 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) + 4 \sin (x) \cos (x) (16 \sin^{4} (x) \cos^{2} (x))$

Etapa 4.5

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

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Etapa 4.5.1

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

$4 \sin (x) \cos (x) + 4 \sin (x) \cos^{3} (x) \cdot 2 + 4 \sin^{3} (x) \cos (x) \cdot - 12 + 4 \sin^{5} (x) \cos (x) \cdot 24 + 4 (\sin^{2} (x) \sin (x)) \cos (x) (- 8 \cos^{2} (x)) + 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) + 4 \sin (x) \cos (x) (16 \sin^{4} (x) \cos^{2} (x))$

Etapa 4.5.2

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

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Etapa 4.5.2.1

Eleve $\sin (x)$ à potência de $1$ .

$4 \sin (x) \cos (x) + 4 \sin (x) \cos^{3} (x) \cdot 2 + 4 \sin^{3} (x) \cos (x) \cdot - 12 + 4 \sin^{5} (x) \cos (x) \cdot 24 + 4 (\sin^{2} (x) \sin^{1} (x)) \cos (x) (- 8 \cos^{2} (x)) + 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) + 4 \sin (x) \cos (x) (16 \sin^{4} (x) \cos^{2} (x))$

Etapa 4.5.2.2

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

$4 \sin (x) \cos (x) + 4 \sin (x) \cos^{3} (x) \cdot 2 + 4 \sin^{3} (x) \cos (x) \cdot - 12 + 4 \sin^{5} (x) \cos (x) \cdot 24 + 4 {\sin (x)}^{2 + 1} \cos (x) (- 8 \cos^{2} (x)) + 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) + 4 \sin (x) \cos (x) (16 \sin^{4} (x) \cos^{2} (x))$

Etapa 4.5.3

Some $2$ e $1$ .

$4 \sin (x) \cos (x) + 4 \sin (x) \cos^{3} (x) \cdot 2 + 4 \sin^{3} (x) \cos (x) \cdot - 12 + 4 \sin^{5} (x) \cos (x) \cdot 24 + 4 \sin^{3} (x) \cos (x) (- 8 \cos^{2} (x)) + 4 \sin (x) \cos (x) (- 16 \sin^{6} (x)) + 4 \sin (x) \cos (x) (16 \sin^{4} (x) \cos^{2} (x))$

Etapa 4.6

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

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Etapa 4.6.1

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

$4 \sin (x) \cos (x) + 4 \sin (x) \cos^{3} (x) \cdot 2 + 4 \sin^{3} (x) \cos (x) \cdot - 12 + 4 \sin^{5} (x) \cos (x) \cdot 24 + 4 \sin^{3} (x) \cos (x) (- 8 \cos^{2} (x)) + 4 (\sin^{6} (x) \sin (x)) \cos (x) \cdot - 16 + 4 \sin (x) \cos (x) (16 \sin^{4} (x) \cos^{2} (x))$

Etapa 4.6.2

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

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Etapa 4.6.2.1

Eleve $\sin (x)$ à potência de $1$ .

$4 \sin (x) \cos (x) + 4 \sin (x) \cos^{3} (x) \cdot 2 + 4 \sin^{3} (x) \cos (x) \cdot - 12 + 4 \sin^{5} (x) \cos (x) \cdot 24 + 4 \sin^{3} (x) \cos (x) (- 8 \cos^{2} (x)) + 4 (\sin^{6} (x) \sin^{1} (x)) \cos (x) \cdot - 16 + 4 \sin (x) \cos (x) (16 \sin^{4} (x) \cos^{2} (x))$

Etapa 4.6.2.2

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

$4 \sin (x) \cos (x) + 4 \sin (x) \cos^{3} (x) \cdot 2 + 4 \sin^{3} (x) \cos (x) \cdot - 12 + 4 \sin^{5} (x) \cos (x) \cdot 24 + 4 \sin^{3} (x) \cos (x) (- 8 \cos^{2} (x)) + 4 {\sin (x)}^{6 + 1} \cos (x) \cdot - 16 + 4 \sin (x) \cos (x) (16 \sin^{4} (x) \cos^{2} (x))$

Etapa 4.6.3

Some $6$ e $1$ .

$4 \sin (x) \cos (x) + 4 \sin (x) \cos^{3} (x) \cdot 2 + 4 \sin^{3} (x) \cos (x) \cdot - 12 + 4 \sin^{5} (x) \cos (x) \cdot 24 + 4 \sin^{3} (x) \cos (x) (- 8 \cos^{2} (x)) + 4 \sin^{7} (x) \cos (x) \cdot - 16 + 4 \sin (x) \cos (x) (16 \sin^{4} (x) \cos^{2} (x))$

Etapa 4.7

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

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Etapa 4.7.1

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

Etapa 4.7.2

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

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Etapa 4.7.2.1

Eleve $\sin (x)$ à potência de $1$ .

Etapa 4.7.2.2

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

Etapa 4.7.3

Some $4$ e $1$ .

Etapa 5

Simplifique cada termo.

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Etapa 5.1

Multiplique $2$ por $4$ .

$4 \sin (x) \cos (x) + 8 \sin (x) \cos^{3} (x) + 4 \sin^{3} (x) \cos (x) \cdot - 12 + 4 \sin^{5} (x) \cos (x) \cdot 24 + 4 \sin^{3} (x) \cos (x) (- 8 \cos^{2} (x)) + 4 \sin^{7} (x) \cos (x) \cdot - 16 + 4 \sin^{5} (x) \cos (x) (16 \cos^{2} (x))$

Etapa 5.2

Multiplique $- 12$ por $4$ .

$4 \sin (x) \cos (x) + 8 \sin (x) \cos^{3} (x) - 48 \sin^{3} (x) \cos (x) + 4 \sin^{5} (x) \cos (x) \cdot 24 + 4 \sin^{3} (x) \cos (x) (- 8 \cos^{2} (x)) + 4 \sin^{7} (x) \cos (x) \cdot - 16 + 4 \sin^{5} (x) \cos (x) (16 \cos^{2} (x))$

Etapa 5.3

Multiplique $24$ por $4$ .

$4 \sin (x) \cos (x) + 8 \sin (x) \cos^{3} (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) + 4 \sin^{3} (x) \cos (x) (- 8 \cos^{2} (x)) + 4 \sin^{7} (x) \cos (x) \cdot - 16 + 4 \sin^{5} (x) \cos (x) (16 \cos^{2} (x))$

Etapa 5.4

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

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Etapa 5.4.1

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

$4 \sin (x) \cos (x) + 8 \sin (x) \cos^{3} (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) + 4 \sin^{3} (x) (\cos^{2} (x) \cos (x)) \cdot - 8 + 4 \sin^{7} (x) \cos (x) \cdot - 16 + 4 \sin^{5} (x) \cos (x) (16 \cos^{2} (x))$

Etapa 5.4.2

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

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Etapa 5.4.2.1

Eleve $\cos (x)$ à potência de $1$ .

$4 \sin (x) \cos (x) + 8 \sin (x) \cos^{3} (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) + 4 \sin^{3} (x) (\cos^{2} (x) \cos^{1} (x)) \cdot - 8 + 4 \sin^{7} (x) \cos (x) \cdot - 16 + 4 \sin^{5} (x) \cos (x) (16 \cos^{2} (x))$

Etapa 5.4.2.2

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

$4 \sin (x) \cos (x) + 8 \sin (x) \cos^{3} (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) + 4 \sin^{3} (x) {\cos (x)}^{2 + 1} \cdot - 8 + 4 \sin^{7} (x) \cos (x) \cdot - 16 + 4 \sin^{5} (x) \cos (x) (16 \cos^{2} (x))$

Etapa 5.4.3

Some $2$ e $1$ .

$4 \sin (x) \cos (x) + 8 \sin (x) \cos^{3} (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) + 4 \sin^{3} (x) \cos^{3} (x) \cdot - 8 + 4 \sin^{7} (x) \cos (x) \cdot - 16 + 4 \sin^{5} (x) \cos (x) (16 \cos^{2} (x))$

Etapa 5.5

Multiplique $- 8$ por $4$ .

$4 \sin (x) \cos (x) + 8 \sin (x) \cos^{3} (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) + 4 \sin^{7} (x) \cos (x) \cdot - 16 + 4 \sin^{5} (x) \cos (x) (16 \cos^{2} (x))$

Etapa 5.6

Multiplique $- 16$ por $4$ .

$4 \sin (x) \cos (x) + 8 \sin (x) \cos^{3} (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 4 \sin^{5} (x) \cos (x) (16 \cos^{2} (x))$

Etapa 5.7

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

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Etapa 5.7.1

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

$4 \sin (x) \cos (x) + 8 \sin (x) \cos^{3} (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 4 \sin^{5} (x) (\cos^{2} (x) \cos (x)) \cdot 16$

Etapa 5.7.2

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

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Etapa 5.7.2.1

Eleve $\cos (x)$ à potência de $1$ .

Etapa 5.7.2.2

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

$4 \sin (x) \cos (x) + 8 \sin (x) \cos^{3} (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 4 \sin^{5} (x) {\cos (x)}^{2 + 1} \cdot 16$

Etapa 5.7.3

Some $2$ e $1$ .

$4 \sin (x) \cos (x) + 8 \sin (x) \cos^{3} (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 4 \sin^{5} (x) \cos^{3} (x) \cdot 16$

Etapa 5.8

Multiplique $16$ por $4$ .

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