Trigonometry Examples

$1 + \cos (12 x)$

Step 1

Simplify each term.

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Step 1.1

Factor $2$ out of $12 x$ .

$1 + \cos (2 (6 x))$

Step 1.2

Simplify each term.

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Step 1.2.1

Factor $2$ out of $6 x$ .

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

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

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

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

Step 1.2.2.2

Rewrite ${(4 \cos^{3} (x) - 3 \cos (x))}^{2}$ as $(4 \cos^{3} (x) - 3 \cos (x)) (4 \cos^{3} (x) - 3 \cos (x))$ .

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

Step 1.2.2.3

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

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Step 1.2.2.3.1

Apply the distributive property.

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

Step 1.2.2.3.2

Apply the distributive property.

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

Step 1.2.2.3.3

Apply the distributive property.

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

Step 1.2.2.4

Simplify and combine like terms.

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Step 1.2.2.4.1

Simplify each term.

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Step 1.2.2.4.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.2.2.4.1.2

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

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Step 1.2.2.4.1.2.1

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

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

Step 1.2.2.4.1.2.2

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

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

Step 1.2.2.4.1.2.3

Add $3$ and $3$ .

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

Step 1.2.2.4.1.3

Multiply $4$ by $4$ .

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

Step 1.2.2.4.1.4

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

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Step 1.2.2.4.1.4.1

Move $\cos (x)$ .

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

Step 1.2.2.4.1.4.2

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

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Step 1.2.2.4.1.4.2.1

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

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

Step 1.2.2.4.1.4.2.2

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

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

Step 1.2.2.4.1.4.3

Add $1$ and $3$ .

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

Step 1.2.2.4.1.5

Multiply $- 3$ by $4$ .

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

Step 1.2.2.4.1.6

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

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Step 1.2.2.4.1.6.1

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

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

Step 1.2.2.4.1.6.2

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

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Step 1.2.2.4.1.6.2.1

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

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

Step 1.2.2.4.1.6.2.2

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

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

Step 1.2.2.4.1.6.3

Add $3$ and $1$ .

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

Step 1.2.2.4.1.7

Multiply $4$ by $- 3$ .

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

Step 1.2.2.4.1.8

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

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Step 1.2.2.4.1.8.1

Multiply $- 3$ by $- 3$ .

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

Step 1.2.2.4.1.8.2

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

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

Step 1.2.2.4.1.8.3

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

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

Step 1.2.2.4.1.8.4

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

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

Step 1.2.2.4.1.8.5

Add $1$ and $1$ .

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

Step 1.2.2.4.2

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

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

Step 1.2.2.5

Apply the distributive property.

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

Step 1.2.2.6

Simplify.

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Step 1.2.2.6.1

Multiply $16$ by $2$ .

$1 + 2 {(32 \cos^{6} (x) + 2 (- 24 \cos^{4} (x)) + 2 (9 \cos^{2} (x)) - 1)}^{2} - 1$

Step 1.2.2.6.2

Multiply $- 24$ by $2$ .

$1 + 2 {(32 \cos^{6} (x) - 48 \cos^{4} (x) + 2 (9 \cos^{2} (x)) - 1)}^{2} - 1$

Step 1.2.2.6.3

Multiply $9$ by $2$ .

$1 + 2 {(32 \cos^{6} (x) - 48 \cos^{4} (x) + 18 \cos^{2} (x) - 1)}^{2} - 1$

Step 1.2.3

Rewrite ${(32 \cos^{6} (x) - 48 \cos^{4} (x) + 18 \cos^{2} (x) - 1)}^{2}$ as $(32 \cos^{6} (x) - 48 \cos^{4} (x) + 18 \cos^{2} (x) - 1) (32 \cos^{6} (x) - 48 \cos^{4} (x) + 18 \cos^{2} (x) - 1)$ .

$1 + 2 ((32 \cos^{6} (x) - 48 \cos^{4} (x) + 18 \cos^{2} (x) - 1) (32 \cos^{6} (x) - 48 \cos^{4} (x) + 18 \cos^{2} (x) - 1)) - 1$

Step 1.2.4

Expand $(32 \cos^{6} (x) - 48 \cos^{4} (x) + 18 \cos^{2} (x) - 1) (32 \cos^{6} (x) - 48 \cos^{4} (x) + 18 \cos^{2} (x) - 1)$ by multiplying each term in the first expression by each term in the second expression.

$1 + 2 (32 \cos^{6} (x) (32 \cos^{6} (x)) + 32 \cos^{6} (x) (- 48 \cos^{4} (x)) + 32 \cos^{6} (x) (18 \cos^{2} (x)) + 32 \cos^{6} (x) \cdot - 1 - 48 \cos^{4} (x) (32 \cos^{6} (x)) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5

Simplify each term.

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Step 1.2.5.1

Rewrite using the commutative property of multiplication.

$1 + 2 (32 \cdot 32 \cos^{6} (x) \cos^{6} (x) + 32 \cos^{6} (x) (- 48 \cos^{4} (x)) + 32 \cos^{6} (x) (18 \cos^{2} (x)) + 32 \cos^{6} (x) \cdot - 1 - 48 \cos^{4} (x) (32 \cos^{6} (x)) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.2

Multiply $\cos^{6} (x)$ by $\cos^{6} (x)$ by adding the exponents.

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Step 1.2.5.2.1

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

$1 + 2 (32 \cdot 32 (\cos^{6} (x) \cos^{6} (x)) + 32 \cos^{6} (x) (- 48 \cos^{4} (x)) + 32 \cos^{6} (x) (18 \cos^{2} (x)) + 32 \cos^{6} (x) \cdot - 1 - 48 \cos^{4} (x) (32 \cos^{6} (x)) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.2.2

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

$1 + 2 (32 \cdot 32 {\cos (x)}^{6 + 6} + 32 \cos^{6} (x) (- 48 \cos^{4} (x)) + 32 \cos^{6} (x) (18 \cos^{2} (x)) + 32 \cos^{6} (x) \cdot - 1 - 48 \cos^{4} (x) (32 \cos^{6} (x)) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.2.3

Add $6$ and $6$ .

$1 + 2 (32 \cdot 32 \cos^{12} (x) + 32 \cos^{6} (x) (- 48 \cos^{4} (x)) + 32 \cos^{6} (x) (18 \cos^{2} (x)) + 32 \cos^{6} (x) \cdot - 1 - 48 \cos^{4} (x) (32 \cos^{6} (x)) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.3

Multiply $32$ by $32$ .

$1 + 2 (1024 \cos^{12} (x) + 32 \cos^{6} (x) (- 48 \cos^{4} (x)) + 32 \cos^{6} (x) (18 \cos^{2} (x)) + 32 \cos^{6} (x) \cdot - 1 - 48 \cos^{4} (x) (32 \cos^{6} (x)) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.4

Rewrite using the commutative property of multiplication.

$1 + 2 (1024 \cos^{12} (x) + 32 \cdot - 48 \cos^{6} (x) \cos^{4} (x) + 32 \cos^{6} (x) (18 \cos^{2} (x)) + 32 \cos^{6} (x) \cdot - 1 - 48 \cos^{4} (x) (32 \cos^{6} (x)) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.5

Multiply $\cos^{6} (x)$ by $\cos^{4} (x)$ by adding the exponents.

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Step 1.2.5.5.1

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

$1 + 2 (1024 \cos^{12} (x) + 32 \cdot - 48 (\cos^{4} (x) \cos^{6} (x)) + 32 \cos^{6} (x) (18 \cos^{2} (x)) + 32 \cos^{6} (x) \cdot - 1 - 48 \cos^{4} (x) (32 \cos^{6} (x)) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.5.2

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

$1 + 2 (1024 \cos^{12} (x) + 32 \cdot - 48 {\cos (x)}^{4 + 6} + 32 \cos^{6} (x) (18 \cos^{2} (x)) + 32 \cos^{6} (x) \cdot - 1 - 48 \cos^{4} (x) (32 \cos^{6} (x)) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.5.3

Add $4$ and $6$ .

$1 + 2 (1024 \cos^{12} (x) + 32 \cdot - 48 \cos^{10} (x) + 32 \cos^{6} (x) (18 \cos^{2} (x)) + 32 \cos^{6} (x) \cdot - 1 - 48 \cos^{4} (x) (32 \cos^{6} (x)) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.6

Multiply $32$ by $- 48$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 32 \cos^{6} (x) (18 \cos^{2} (x)) + 32 \cos^{6} (x) \cdot - 1 - 48 \cos^{4} (x) (32 \cos^{6} (x)) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.7

Rewrite using the commutative property of multiplication.

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 32 \cdot 18 \cos^{6} (x) \cos^{2} (x) + 32 \cos^{6} (x) \cdot - 1 - 48 \cos^{4} (x) (32 \cos^{6} (x)) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.8

Multiply $\cos^{6} (x)$ by $\cos^{2} (x)$ by adding the exponents.

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Step 1.2.5.8.1

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

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 32 \cdot 18 (\cos^{2} (x) \cos^{6} (x)) + 32 \cos^{6} (x) \cdot - 1 - 48 \cos^{4} (x) (32 \cos^{6} (x)) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.8.2

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

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 32 \cdot 18 {\cos (x)}^{2 + 6} + 32 \cos^{6} (x) \cdot - 1 - 48 \cos^{4} (x) (32 \cos^{6} (x)) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.8.3

Add $2$ and $6$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 32 \cdot 18 \cos^{8} (x) + 32 \cos^{6} (x) \cdot - 1 - 48 \cos^{4} (x) (32 \cos^{6} (x)) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.9

Multiply $32$ by $18$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) + 32 \cos^{6} (x) \cdot - 1 - 48 \cos^{4} (x) (32 \cos^{6} (x)) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.10

Multiply $- 1$ by $32$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 48 \cos^{4} (x) (32 \cos^{6} (x)) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.11

Rewrite using the commutative property of multiplication.

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 48 \cdot 32 \cos^{4} (x) \cos^{6} (x) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.12

Multiply $\cos^{4} (x)$ by $\cos^{6} (x)$ by adding the exponents.

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Step 1.2.5.12.1

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

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 48 \cdot 32 (\cos^{6} (x) \cos^{4} (x)) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.12.2

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

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 48 \cdot 32 {\cos (x)}^{6 + 4} - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.12.3

Add $6$ and $4$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 48 \cdot 32 \cos^{10} (x) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.13

Multiply $- 48$ by $32$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) - 48 \cos^{4} (x) (- 48 \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.14

Rewrite using the commutative property of multiplication.

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) - 48 \cdot - 48 \cos^{4} (x) \cos^{4} (x) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.15

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

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Step 1.2.5.15.1

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

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) - 48 \cdot - 48 (\cos^{4} (x) \cos^{4} (x)) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.15.2

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

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) - 48 \cdot - 48 {\cos (x)}^{4 + 4} - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.15.3

Add $4$ and $4$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) - 48 \cdot - 48 \cos^{8} (x) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.16

Multiply $- 48$ by $- 48$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 48 \cos^{4} (x) (18 \cos^{2} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.17

Rewrite using the commutative property of multiplication.

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 48 \cdot 18 \cos^{4} (x) \cos^{2} (x) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.18

Multiply $\cos^{4} (x)$ by $\cos^{2} (x)$ by adding the exponents.

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Step 1.2.5.18.1

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

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 48 \cdot 18 (\cos^{2} (x) \cos^{4} (x)) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.18.2

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

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 48 \cdot 18 {\cos (x)}^{2 + 4} - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.18.3

Add $2$ and $4$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 48 \cdot 18 \cos^{6} (x) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.19

Multiply $- 48$ by $18$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) - 48 \cos^{4} (x) \cdot - 1 + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.20

Multiply $- 1$ by $- 48$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 18 \cos^{2} (x) (32 \cos^{6} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.21

Rewrite using the commutative property of multiplication.

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 18 \cdot 32 \cos^{2} (x) \cos^{6} (x) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.22

Multiply $\cos^{2} (x)$ by $\cos^{6} (x)$ by adding the exponents.

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Step 1.2.5.22.1

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

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 18 \cdot 32 (\cos^{6} (x) \cos^{2} (x)) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.22.2

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

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 18 \cdot 32 {\cos (x)}^{6 + 2} + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.22.3

Add $6$ and $2$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 18 \cdot 32 \cos^{8} (x) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.23

Multiply $18$ by $32$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 576 \cos^{8} (x) + 18 \cos^{2} (x) (- 48 \cos^{4} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.24

Rewrite using the commutative property of multiplication.

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 576 \cos^{8} (x) + 18 \cdot - 48 \cos^{2} (x) \cos^{4} (x) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.25

Multiply $\cos^{2} (x)$ by $\cos^{4} (x)$ by adding the exponents.

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Step 1.2.5.25.1

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

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 576 \cos^{8} (x) + 18 \cdot - 48 (\cos^{4} (x) \cos^{2} (x)) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.25.2

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

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 576 \cos^{8} (x) + 18 \cdot - 48 {\cos (x)}^{4 + 2} + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.25.3

Add $4$ and $2$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 576 \cos^{8} (x) + 18 \cdot - 48 \cos^{6} (x) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.26

Multiply $18$ by $- 48$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 576 \cos^{8} (x) - 864 \cos^{6} (x) + 18 \cos^{2} (x) (18 \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.27

Rewrite using the commutative property of multiplication.

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 576 \cos^{8} (x) - 864 \cos^{6} (x) + 18 \cdot 18 \cos^{2} (x) \cos^{2} (x) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.28

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

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Step 1.2.5.28.1

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

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 576 \cos^{8} (x) - 864 \cos^{6} (x) + 18 \cdot 18 (\cos^{2} (x) \cos^{2} (x)) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.28.2

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

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 576 \cos^{8} (x) - 864 \cos^{6} (x) + 18 \cdot 18 {\cos (x)}^{2 + 2} + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.28.3

Add $2$ and $2$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 576 \cos^{8} (x) - 864 \cos^{6} (x) + 18 \cdot 18 \cos^{4} (x) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.29

Multiply $18$ by $18$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 576 \cos^{8} (x) - 864 \cos^{6} (x) + 324 \cos^{4} (x) + 18 \cos^{2} (x) \cdot - 1 - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.30

Multiply $- 1$ by $18$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 576 \cos^{8} (x) - 864 \cos^{6} (x) + 324 \cos^{4} (x) - 18 \cos^{2} (x) - 1 (32 \cos^{6} (x)) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.31

Multiply $32$ by $- 1$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 576 \cos^{8} (x) - 864 \cos^{6} (x) + 324 \cos^{4} (x) - 18 \cos^{2} (x) - 32 \cos^{6} (x) - 1 (- 48 \cos^{4} (x)) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.32

Multiply $- 48$ by $- 1$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 576 \cos^{8} (x) - 864 \cos^{6} (x) + 324 \cos^{4} (x) - 18 \cos^{2} (x) - 32 \cos^{6} (x) + 48 \cos^{4} (x) - 1 (18 \cos^{2} (x)) - 1 \cdot - 1) - 1$

Step 1.2.5.33

Multiply $18$ by $- 1$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 576 \cos^{8} (x) - 864 \cos^{6} (x) + 324 \cos^{4} (x) - 18 \cos^{2} (x) - 32 \cos^{6} (x) + 48 \cos^{4} (x) - 18 \cos^{2} (x) - 1 \cdot - 1) - 1$

Step 1.2.5.34

Multiply $- 1$ by $- 1$ .

$1 + 2 (1024 \cos^{12} (x) - 1536 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) - 1536 \cos^{10} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 576 \cos^{8} (x) - 864 \cos^{6} (x) + 324 \cos^{4} (x) - 18 \cos^{2} (x) - 32 \cos^{6} (x) + 48 \cos^{4} (x) - 18 \cos^{2} (x) + 1) - 1$

Step 1.2.6

Subtract $1536 \cos^{10} (x)$ from $- 1536 \cos^{10} (x)$ .

$1 + 2 (1024 \cos^{12} (x) - 3072 \cos^{10} (x) + 576 \cos^{8} (x) - 32 \cos^{6} (x) + 2304 \cos^{8} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 576 \cos^{8} (x) - 864 \cos^{6} (x) + 324 \cos^{4} (x) - 18 \cos^{2} (x) - 32 \cos^{6} (x) + 48 \cos^{4} (x) - 18 \cos^{2} (x) + 1) - 1$

Step 1.2.7

Add $576 \cos^{8} (x)$ and $2304 \cos^{8} (x)$ .

$1 + 2 (1024 \cos^{12} (x) - 3072 \cos^{10} (x) + 2880 \cos^{8} (x) - 32 \cos^{6} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) + 576 \cos^{8} (x) - 864 \cos^{6} (x) + 324 \cos^{4} (x) - 18 \cos^{2} (x) - 32 \cos^{6} (x) + 48 \cos^{4} (x) - 18 \cos^{2} (x) + 1) - 1$

Step 1.2.8

Add $2880 \cos^{8} (x)$ and $576 \cos^{8} (x)$ .

$1 + 2 (1024 \cos^{12} (x) - 3072 \cos^{10} (x) + 3456 \cos^{8} (x) - 32 \cos^{6} (x) - 864 \cos^{6} (x) + 48 \cos^{4} (x) - 864 \cos^{6} (x) + 324 \cos^{4} (x) - 18 \cos^{2} (x) - 32 \cos^{6} (x) + 48 \cos^{4} (x) - 18 \cos^{2} (x) + 1) - 1$

Step 1.2.9

Subtract $864 \cos^{6} (x)$ from $- 32 \cos^{6} (x)$ .

$1 + 2 (1024 \cos^{12} (x) - 3072 \cos^{10} (x) + 3456 \cos^{8} (x) - 896 \cos^{6} (x) + 48 \cos^{4} (x) - 864 \cos^{6} (x) + 324 \cos^{4} (x) - 18 \cos^{2} (x) - 32 \cos^{6} (x) + 48 \cos^{4} (x) - 18 \cos^{2} (x) + 1) - 1$

Step 1.2.10

Subtract $864 \cos^{6} (x)$ from $- 896 \cos^{6} (x)$ .

$1 + 2 (1024 \cos^{12} (x) - 3072 \cos^{10} (x) + 3456 \cos^{8} (x) - 1760 \cos^{6} (x) + 48 \cos^{4} (x) + 324 \cos^{4} (x) - 18 \cos^{2} (x) - 32 \cos^{6} (x) + 48 \cos^{4} (x) - 18 \cos^{2} (x) + 1) - 1$

Step 1.2.11

Subtract $32 \cos^{6} (x)$ from $- 1760 \cos^{6} (x)$ .

$1 + 2 (1024 \cos^{12} (x) - 3072 \cos^{10} (x) + 3456 \cos^{8} (x) - 1792 \cos^{6} (x) + 48 \cos^{4} (x) + 324 \cos^{4} (x) - 18 \cos^{2} (x) + 48 \cos^{4} (x) - 18 \cos^{2} (x) + 1) - 1$

Step 1.2.12

Add $48 \cos^{4} (x)$ and $324 \cos^{4} (x)$ .

$1 + 2 (1024 \cos^{12} (x) - 3072 \cos^{10} (x) + 3456 \cos^{8} (x) - 1792 \cos^{6} (x) + 372 \cos^{4} (x) - 18 \cos^{2} (x) + 48 \cos^{4} (x) - 18 \cos^{2} (x) + 1) - 1$

Step 1.2.13

Add $372 \cos^{4} (x)$ and $48 \cos^{4} (x)$ .

$1 + 2 (1024 \cos^{12} (x) - 3072 \cos^{10} (x) + 3456 \cos^{8} (x) - 1792 \cos^{6} (x) + 420 \cos^{4} (x) - 18 \cos^{2} (x) - 18 \cos^{2} (x) + 1) - 1$

Step 1.2.14

Subtract $18 \cos^{2} (x)$ from $- 18 \cos^{2} (x)$ .

$1 + 2 (1024 \cos^{12} (x) - 3072 \cos^{10} (x) + 3456 \cos^{8} (x) - 1792 \cos^{6} (x) + 420 \cos^{4} (x) - 36 \cos^{2} (x) + 1) - 1$

Step 1.2.15

Apply the distributive property.

$1 + 2 (1024 \cos^{12} (x)) + 2 (- 3072 \cos^{10} (x)) + 2 (3456 \cos^{8} (x)) + 2 (- 1792 \cos^{6} (x)) + 2 (420 \cos^{4} (x)) + 2 (- 36 \cos^{2} (x)) + 2 \cdot 1 - 1$

Step 1.2.16

Simplify.

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Step 1.2.16.1

Multiply $1024$ by $2$ .

$1 + 2048 \cos^{12} (x) + 2 (- 3072 \cos^{10} (x)) + 2 (3456 \cos^{8} (x)) + 2 (- 1792 \cos^{6} (x)) + 2 (420 \cos^{4} (x)) + 2 (- 36 \cos^{2} (x)) + 2 \cdot 1 - 1$

Step 1.2.16.2

Multiply $- 3072$ by $2$ .

$1 + 2048 \cos^{12} (x) - 6144 \cos^{10} (x) + 2 (3456 \cos^{8} (x)) + 2 (- 1792 \cos^{6} (x)) + 2 (420 \cos^{4} (x)) + 2 (- 36 \cos^{2} (x)) + 2 \cdot 1 - 1$

Step 1.2.16.3

Multiply $3456$ by $2$ .

$1 + 2048 \cos^{12} (x) - 6144 \cos^{10} (x) + 6912 \cos^{8} (x) + 2 (- 1792 \cos^{6} (x)) + 2 (420 \cos^{4} (x)) + 2 (- 36 \cos^{2} (x)) + 2 \cdot 1 - 1$

Step 1.2.16.4

Multiply $- 1792$ by $2$ .

$1 + 2048 \cos^{12} (x) - 6144 \cos^{10} (x) + 6912 \cos^{8} (x) - 3584 \cos^{6} (x) + 2 (420 \cos^{4} (x)) + 2 (- 36 \cos^{2} (x)) + 2 \cdot 1 - 1$

Step 1.2.16.5

Multiply $420$ by $2$ .

$1 + 2048 \cos^{12} (x) - 6144 \cos^{10} (x) + 6912 \cos^{8} (x) - 3584 \cos^{6} (x) + 840 \cos^{4} (x) + 2 (- 36 \cos^{2} (x)) + 2 \cdot 1 - 1$

Step 1.2.16.6

Multiply $- 36$ by $2$ .

$1 + 2048 \cos^{12} (x) - 6144 \cos^{10} (x) + 6912 \cos^{8} (x) - 3584 \cos^{6} (x) + 840 \cos^{4} (x) - 72 \cos^{2} (x) + 2 \cdot 1 - 1$

Step 1.2.16.7

Multiply $2$ by $1$ .

$1 + 2048 \cos^{12} (x) - 6144 \cos^{10} (x) + 6912 \cos^{8} (x) - 3584 \cos^{6} (x) + 840 \cos^{4} (x) - 72 \cos^{2} (x) + 2 - 1$

Step 1.3

Subtract $1$ from $2$ .

$1 + 2048 \cos^{12} (x) - 6144 \cos^{10} (x) + 6912 \cos^{8} (x) - 3584 \cos^{6} (x) + 840 \cos^{4} (x) - 72 \cos^{2} (x) + 1$

Step 2

Add $1$ and $1$ .

$2 + 2048 \cos^{12} (x) - 6144 \cos^{10} (x) + 6912 \cos^{8} (x) - 3584 \cos^{6} (x) + 840 \cos^{4} (x) - 72 \cos^{2} (x)$