Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Factor $2$ out of $8 x$ .

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

Step 1.2

Factor $2$ out of $4 x$ .

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

Step 1.3

Apply the sine double-angle identity.

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

Step 1.4

Multiply $2$ by $2$ .

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

Step 1.5

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

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

Step 1.6

Apply the distributive property.

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

Step 1.7

Multiply $4$ by $1$ .

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

Step 1.8

Multiply $\sin (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 1.8.1

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

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

Step 1.8.2

Multiply $\sin^{2} (x)$ by $\sin (x)$ .

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Step 1.8.2.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.8.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.8.3

Add $2$ and $1$ .

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

Step 1.9

Multiply $- 2$ by $4$ .

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

Step 1.10

Apply the distributive property.

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

Step 1.11

Multiply $4$ by $2$ .

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

Step 1.12

Multiply $- 8$ by $2$ .

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

Step 1.13

Remove parentheses.

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

Step 1.14

Factor $2$ out of $4 x$ .

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

Step 1.15

Simplify each term.

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Step 1.15.1

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

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

Step 1.15.2

Rewrite ${(1 - 2 \sin^{2} (x))}^{2}$ as $(1 - 2 \sin^{2} (x)) (1 - 2 \sin^{2} (x))$ .

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

Step 1.15.3

Expand $(1 - 2 \sin^{2} (x)) (1 - 2 \sin^{2} (x))$ using the FOIL Method.

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Step 1.15.3.1

Apply the distributive property.

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

Step 1.15.3.2

Apply the distributive property.

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

Step 1.15.3.3

Apply the distributive property.

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

Step 1.15.4

Simplify and combine like terms.

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Step 1.15.4.1

Simplify each term.

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Step 1.15.4.1.1

Multiply $1$ by $1$ .

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

Step 1.15.4.1.2

Multiply $- 2 \sin^{2} (x)$ by $1$ .

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

Step 1.15.4.1.3

Multiply $- 2$ by $1$ .

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

Step 1.15.4.1.4

Rewrite using the commutative property of multiplication.

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

Step 1.15.4.1.5

Multiply $\sin^{2} (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 1.15.4.1.5.1

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

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

Step 1.15.4.1.5.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.15.4.1.5.3

Add $2$ and $2$ .

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

Step 1.15.4.1.6

Multiply $- 2$ by $- 2$ .

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

Step 1.15.4.2

Subtract $2 \sin^{2} (x)$ from $- 2 \sin^{2} (x)$ .

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

Step 1.15.5

Apply the sine double-angle identity.

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

Step 1.15.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.15.6.1

Apply the product rule to $2 \sin (x) \cos (x)$ .

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

Step 1.15.6.2

Apply the product rule to $2 \sin (x)$ .

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

Step 1.15.7

Raise $2$ to the power of $2$ .

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

Step 1.15.8

Multiply $4$ by $- 1$ .

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

Step 1.16

Expand $(8 \sin (x) \cos (x) - 16 \sin^{3} (x) \cos (x)) (1 - 4 \sin^{2} (x) + 4 \sin^{4} (x) - 4 \sin^{2} (x) \cos^{2} (x))$ by multiplying each term in the first expression by each term in the second expression.

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

Step 1.17

Simplify each term.

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Step 1.17.1

Multiply $8$ by $1$ .

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

Step 1.17.2

Multiply $\sin (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 1.17.2.1

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

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

Step 1.17.2.2

Multiply $\sin^{2} (x)$ by $\sin (x)$ .

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Step 1.17.2.2.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.17.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.17.2.3

Add $2$ and $1$ .

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

Step 1.17.3

Multiply $- 4$ by $8$ .

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

Step 1.17.4

Multiply $\sin (x)$ by $\sin^{4} (x)$ by adding the exponents.

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Step 1.17.4.1

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

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

Step 1.17.4.2

Multiply $\sin^{4} (x)$ by $\sin (x)$ .

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Step 1.17.4.2.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.17.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.17.4.3

Add $4$ and $1$ .

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

Step 1.17.5

Multiply $4$ by $8$ .

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

Step 1.17.6

Multiply $\sin (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 1.17.6.1

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

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

Step 1.17.6.2

Multiply $\sin^{2} (x)$ by $\sin (x)$ .

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Step 1.17.6.2.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.17.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.17.6.3

Add $2$ and $1$ .

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

Step 1.17.7

Multiply $\cos (x)$ by $\cos^{2} (x)$ by adding the exponents.

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Step 1.17.7.1

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

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

Step 1.17.7.2

Multiply $\cos^{2} (x)$ by $\cos (x)$ .

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Step 1.17.7.2.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 1.17.7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.17.7.3

Add $2$ and $1$ .

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

Step 1.17.8

Multiply $- 4$ by $8$ .

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

Step 1.17.9

Multiply $- 16$ by $1$ .

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

Step 1.17.10

Multiply $\sin^{3} (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 1.17.10.1

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

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

Step 1.17.10.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.17.10.3

Add $2$ and $3$ .

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

Step 1.17.11

Multiply $- 4$ by $- 16$ .

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

Step 1.17.12

Multiply $\sin^{3} (x)$ by $\sin^{4} (x)$ by adding the exponents.

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Step 1.17.12.1

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

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

Step 1.17.12.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.17.12.3

Add $4$ and $3$ .

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

Step 1.17.13

Multiply $4$ by $- 16$ .

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

Step 1.17.14

Multiply $\sin^{3} (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 1.17.14.1

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

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

Step 1.17.14.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.17.14.3

Add $2$ and $3$ .

Step 1.17.15

Multiply $\cos (x)$ by $\cos^{2} (x)$ by adding the exponents.

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Step 1.17.15.1

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

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

Step 1.17.15.2

Multiply $\cos^{2} (x)$ by $\cos (x)$ .

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Step 1.17.15.2.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 1.17.15.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

Step 1.17.15.3

Add $2$ and $1$ .

Step 1.17.16

Multiply $- 4$ by $- 16$ .

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

Step 1.18

Subtract $16 \sin^{3} (x) \cos (x)$ from $- 32 \sin^{3} (x) \cos (x)$ .

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

Step 1.19

Add $32 \sin^{5} (x) \cos (x)$ and $64 \sin^{5} (x) \cos (x)$ .

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

Step 1.20

Factor $2$ out of $6 x$ .

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

Step 1.21

Apply the sine triple-angle identity.

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

Step 1.22

Apply the distributive property.

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

Step 1.23

Multiply $- 4$ by $2$ .

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

Step 1.24

Multiply $3$ by $2$ .

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

Step 1.25

Use the triple-angle identity to transform $\cos (3 x)$ to $4 \cos^{3} (x) - 3 \cos (x)$ .

Step 1.26

Expand $(- 8 \sin^{3} (x) + 6 \sin (x)) (4 \cos^{3} (x) - 3 \cos (x))$ using the FOIL Method.

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Step 1.26.1

Apply the distributive property.

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

Step 1.26.2

Apply the distributive property.

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

Step 1.26.3

Apply the distributive property.

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

Step 1.27

Simplify each term.

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Step 1.27.1

Rewrite using the commutative property of multiplication.

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

Step 1.27.2

Multiply $- 8$ by $4$ .

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

Step 1.27.3

Multiply $- 3$ by $- 8$ .

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

Step 1.27.4

Multiply $4$ by $6$ .

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

Step 1.27.5

Multiply $- 3$ by $6$ .

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

Step 1.28

Factor $2 \sin (x) \cos (x)$ out of $- 32 \sin^{3} (x) \cos^{3} (x) + 24 \sin^{3} (x) \cos (x) + 24 \sin (x) \cos^{3} (x) - 18 \sin (x) \cos (x)$ .

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Step 1.28.1

Factor $2 \sin (x) \cos (x)$ out of $- 32 \sin^{3} (x) \cos^{3} (x)$ .

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

Step 1.28.2

Factor $2 \sin (x) \cos (x)$ out of $24 \sin^{3} (x) \cos (x)$ .

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

Step 1.28.3

Factor $2 \sin (x) \cos (x)$ out of $24 \sin (x) \cos^{3} (x)$ .

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

Step 1.28.4

Factor $2 \sin (x) \cos (x)$ out of $- 18 \sin (x) \cos (x)$ .

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

Step 1.28.5

Factor $2 \sin (x) \cos (x)$ out of $2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x)) + 2 \sin (x) \cos (x) (12 \sin^{2} (x))$ .

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

Step 1.28.6

Factor $2 \sin (x) \cos (x)$ out of $2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x) + 12 \sin^{2} (x)) + 2 \sin (x) \cos (x) (12 \cos^{2} (x))$ .

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

Step 1.28.7

Factor $2 \sin (x) \cos (x)$ out of $2 \sin (x) \cos (x) (- 16 \sin^{2} (x) \cos^{2} (x) + 12 \sin^{2} (x) + 12 \cos^{2} (x)) + 2 \sin (x) \cos (x) (- 9)$ .

Step 1.29

Factor $12$ out of $12 \sin^{2} (x)$ .

Step 1.30

Factor $12$ out of $12 \cos^{2} (x)$ .

Step 1.31

Factor $12$ out of $12 (\sin^{2} (x)) + 12 (\cos^{2} (x))$ .

Step 1.32

Apply pythagorean identity.

Step 1.33

Multiply $12$ by $1$ .

Step 1.34

Subtract $9$ from $12$ .

Step 1.35

Apply the distributive property.

Step 1.36

Multiply $\sin (x)$ by $\sin^{2} (x)$ by adding the exponents.

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Step 1.36.1

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

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

Step 1.36.2

Multiply $\sin^{2} (x)$ by $\sin (x)$ .

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Step 1.36.2.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.36.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.36.3

Add $2$ and $1$ .

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

Step 1.37

Multiply $3$ by $2$ .

Step 1.38

Simplify each term.

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Step 1.38.1

Multiply $\cos (x)$ by $\cos^{2} (x)$ by adding the exponents.

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Step 1.38.1.1

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

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

Step 1.38.1.2

Multiply $\cos^{2} (x)$ by $\cos (x)$ .

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Step 1.38.1.2.1

Raise $\cos (x)$ to the power of $1$ .

Step 1.38.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.38.1.3

Add $2$ and $1$ .

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

Step 1.38.2

Multiply $- 16$ by $2$ .

Step 1.39

Apply the distributive property.

Step 1.40

Multiply $- 32$ by $- 1$ .

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

Step 1.41

Multiply $6$ by $- 1$ .

Step 1.42

Remove parentheses.

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

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $8 \sin (x) \cos (x) - 48 \sin^{3} (x) \cos (x) + 96 \sin^{5} (x) \cos (x) - 32 \sin^{3} (x) \cos^{3} (x) - 64 \sin^{7} (x) \cos (x) + 64 \sin^{5} (x) \cos^{3} (x) + 32 \sin^{3} (x) \cos^{3} (x) - 6 \sin (x) \cos (x)$ .

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Step 2.1.1

Add $- 32 \sin^{3} (x) \cos^{3} (x)$ and $32 \sin^{3} (x) \cos^{3} (x)$ .

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

Step 2.1.2

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

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

Step 2.2

Subtract $6 \sin (x) \cos (x)$ from $8 \sin (x) \cos (x)$ .

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