Trigonometry Examples

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

Step 1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (3 x)$ and $b = \sin (3 x)$ .

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

Apply the sine triple-angle identity.

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

Step 2.2

Simplify each term.

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Step 2.2.1

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

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

Step 2.2.2

Apply the sine triple-angle identity.

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

Step 2.2.3

Apply the distributive property.

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

Step 2.2.4

Multiply $- 4$ by $- 1$ .

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

Step 2.2.5

Multiply $3$ by $- 1$ .

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

Step 3

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

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

Step 4

Simplify terms.

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Step 4.1

Combine the opposite terms in $4 \cos^{3} (x) (4 \cos^{3} (x)) + 4 \cos^{3} (x) (- 3 \cos (x)) + 4 \cos^{3} (x) (4 \sin^{3} (x)) + 4 \cos^{3} (x) (- 3 \sin (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x)) - 3 \cos (x) (4 \sin^{3} (x)) - 3 \cos (x) (- 3 \sin (x)) - 4 \sin^{3} (x) (4 \cos^{3} (x)) - 4 \sin^{3} (x) (- 3 \cos (x)) - 4 \sin^{3} (x) (4 \sin^{3} (x)) - 4 \sin^{3} (x) (- 3 \sin (x)) + 3 \sin (x) (4 \cos^{3} (x)) + 3 \sin (x) (- 3 \cos (x)) + 3 \sin (x) (4 \sin^{3} (x)) + 3 \sin (x) (- 3 \sin (x))$ .

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Step 4.1.1

Reorder the factors in the terms $4 \cos^{3} (x) (4 \sin^{3} (x))$ and $- 4 \sin^{3} (x) (4 \cos^{3} (x))$ .

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

Step 4.1.2

Subtract $4 \cdot 4 \cos^{3} (x) \sin^{3} (x)$ from $4 \cdot 4 \cos^{3} (x) \sin^{3} (x)$ .

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

Step 4.1.3

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

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

Step 4.1.4

Reorder the factors in the terms $4 \cos^{3} (x) (- 3 \sin (x))$ and $3 \sin (x) (4 \cos^{3} (x))$ .

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

Step 4.1.5

Add $- 3 \cdot 4 \cos^{3} (x) \sin (x)$ and $3 \cdot 4 \cos^{3} (x) \sin (x)$ .

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

Step 4.1.6

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

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

Step 4.2

Simplify each term.

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Step 4.2.1

Rewrite using the commutative property of multiplication.

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

Step 4.2.2

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

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

Step 4.2.2.3

Add $3$ and $3$ .

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

Step 4.2.3

Multiply $4$ by $4$ .

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

Step 4.2.4

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

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Step 4.2.4.1

Move $\cos (x)$ .

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

Step 4.2.4.2

Multiply $\cos (x)$ by $\cos^{3} (x)$ .

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Step 4.2.4.2.1

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

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

Step 4.2.4.2.2

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

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

Step 4.2.4.3

Add $1$ and $3$ .

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

Step 4.2.5

Multiply $- 3$ by $4$ .

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

Step 4.2.6

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

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Step 4.2.6.1

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

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

Step 4.2.6.2

Multiply $\cos^{3} (x)$ by $\cos (x)$ .

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Step 4.2.6.2.1

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

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

Step 4.2.6.2.2

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

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

Step 4.2.6.3

Add $3$ and $1$ .

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

Step 4.2.7

Multiply $4$ by $- 3$ .

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

Step 4.2.8

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

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Step 4.2.8.1

Multiply $- 3$ by $- 3$ .

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

Step 4.2.8.2

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

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

Step 4.2.8.3

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

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

Step 4.2.8.4

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

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

Step 4.2.8.5

Add $1$ and $1$ .

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

Step 4.2.9

Multiply $4$ by $- 3$ .

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

Step 4.2.10

Multiply $- 3$ by $- 3$ .

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

Step 4.2.11

Multiply $- 3$ by $- 4$ .

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

Step 4.2.12

Rewrite using the commutative property of multiplication.

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

Step 4.2.13

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

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Step 4.2.13.1

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

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

Step 4.2.13.2

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

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

Step 4.2.13.3

Add $3$ and $3$ .

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

Step 4.2.14

Multiply $- 4$ by $4$ .

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

Step 4.2.15

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

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Step 4.2.15.1

Move $\sin (x)$ .

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

Step 4.2.15.2

Multiply $\sin (x)$ by $\sin^{3} (x)$ .

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Step 4.2.15.2.1

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

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

Step 4.2.15.2.2

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

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

Step 4.2.15.3

Add $1$ and $3$ .

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

Step 4.2.16

Multiply $- 3$ by $- 4$ .

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

Step 4.2.17

Multiply $- 3$ by $3$ .

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

Step 4.2.18

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

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Step 4.2.18.1

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

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

Step 4.2.18.2

Multiply $\sin^{3} (x)$ by $\sin (x)$ .

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Step 4.2.18.2.1

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

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

Step 4.2.18.2.2

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

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

Step 4.2.18.3

Add $3$ and $1$ .

Step 4.2.19

Multiply $4$ by $3$ .

Step 4.2.20

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

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Step 4.2.20.1

Multiply $- 3$ by $3$ .

Step 4.2.20.2

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

Step 4.2.20.3

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

Step 4.2.20.4

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

Step 4.2.20.5

Add $1$ and $1$ .

Step 4.3

Simplify by adding terms.

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Step 4.3.1

Combine the opposite terms in $16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x) - 12 \cos (x) \sin^{3} (x) + 9 \cos (x) \sin (x) + 12 \sin^{3} (x) \cos (x) - 16 \sin^{6} (x) + 12 \sin^{4} (x) - 9 \sin (x) \cos (x) + 12 \sin^{4} (x) - 9 \sin^{2} (x)$ .

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Step 4.3.1.1

Reorder the factors in the terms $- 12 \cos (x) \sin^{3} (x)$ and $12 \sin^{3} (x) \cos (x)$ .

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

Step 4.3.1.2

Add $- 12 \sin^{3} (x) \cos (x)$ and $12 \sin^{3} (x) \cos (x)$ .

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

Step 4.3.1.3

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

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

Step 4.3.1.4

Reorder the factors in the terms $9 \cos (x) \sin (x)$ and $- 9 \sin (x) \cos (x)$ .

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

Step 4.3.1.5

Subtract $9 \cos (x) \sin (x)$ from $9 \cos (x) \sin (x)$ .

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

Step 4.3.1.6

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

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

Step 4.3.2

Subtract $12 \cos^{4} (x)$ from $- 12 \cos^{4} (x)$ .

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

Step 4.3.3

Add $12 \sin^{4} (x)$ and $12 \sin^{4} (x)$ .

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