Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Expand $(- 4 \cos (x) \sin (x) + 2 \cos (2 x)) (- 4 \cos (x) \sin (x) + 2 \cos (2 x))$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $- 4 \cos (x) \sin (x) (- 4 \cos (x) \sin (x))$ .

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Step 1.3.1.1.1

Multiply $- 4$ by $- 4$ .

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

Step 1.3.1.1.2

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

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

Step 1.3.1.1.3

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

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

Step 1.3.1.1.4

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

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

Step 1.3.1.1.5

Add $1$ and $1$ .

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

Step 1.3.1.1.6

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

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

Step 1.3.1.1.7

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

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

Step 1.3.1.1.8

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

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

Step 1.3.1.1.9

Add $1$ and $1$ .

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

Step 1.3.1.2

Add parentheses.

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

Step 1.3.1.3

Add parentheses.

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

Step 1.3.1.4

Reorder $\cos (x)$ and $\sin (x) \cdot 2$ .

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

Step 1.3.1.5

Reorder $\sin (x)$ and $2$ .

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

Step 1.3.1.6

Apply the sine double-angle identity.

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

Step 1.3.1.7

Reorder $2 \cos (2 x)$ and $- 4 \cos (x) \sin (x)$ .

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

Step 1.3.1.8

Add parentheses.

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

Step 1.3.1.9

Add parentheses.

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

Step 1.3.1.10

Reorder $\cos (x)$ and $\sin (x) \cdot 2$ .

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

Step 1.3.1.11

Reorder $\sin (x)$ and $2$ .

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

Step 1.3.1.12

Apply the sine double-angle identity.

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

Step 1.3.1.13

Multiply $2 \cos (2 x) (2 \cos (2 x))$ .

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Step 1.3.1.13.1

Multiply $2$ by $2$ .

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

Step 1.3.1.13.2

Raise $\cos (2 x)$ to the power of $1$ .

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

Step 1.3.1.13.3

Raise $\cos (2 x)$ to the power of $1$ .

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

Step 1.3.1.13.4

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

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

Step 1.3.1.13.5

Add $1$ and $1$ .

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

Step 1.3.2

Subtract $4 \sin (2 x) \cos (2 x)$ from $- 4 \sin (2 x) \cos (2 x)$ .

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

Step 1.4

Rewrite ${(2 \cos (2 x) + 4 \sin (x) \cos (x))}^{2}$ as $(2 \cos (2 x) + 4 \sin (x) \cos (x)) (2 \cos (2 x) + 4 \sin (x) \cos (x))$ .

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

Step 1.5

Expand $(2 \cos (2 x) + 4 \sin (x) \cos (x)) (2 \cos (2 x) + 4 \sin (x) \cos (x))$ using the FOIL Method.

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Apply the distributive property.

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

Step 1.5.3

Apply the distributive property.

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

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $2 \cos (2 x) (2 \cos (2 x))$ .

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Step 1.6.1.1.1

Multiply $2$ by $2$ .

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

Step 1.6.1.1.2

Raise $\cos (2 x)$ to the power of $1$ .

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

Step 1.6.1.1.3

Raise $\cos (2 x)$ to the power of $1$ .

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

Step 1.6.1.1.4

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

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

Step 1.6.1.1.5

Add $1$ and $1$ .

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

Step 1.6.1.2

Reorder $2 \cos (2 x)$ and $4 \sin (x) \cos (x)$ .

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

Step 1.6.1.3

Add parentheses.

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

Step 1.6.1.4

Add parentheses.

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

Step 1.6.1.5

Reorder $\sin (x) \cos (x)$ and $2$ .

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

Step 1.6.1.6

Apply the sine double-angle identity.

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

Step 1.6.1.7

Add parentheses.

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

Step 1.6.1.8

Add parentheses.

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

Step 1.6.1.9

Reorder $\sin (x) \cos (x)$ and $2$ .

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

Step 1.6.1.10

Apply the sine double-angle identity.

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

Step 1.6.1.11

Multiply $4 \sin (x) \cos (x) (4 \sin (x) \cos (x))$ .

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Step 1.6.1.11.1

Multiply $4$ by $4$ .

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

Step 1.6.1.11.2

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

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

Step 1.6.1.11.3

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

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

Step 1.6.1.11.4

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

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

Step 1.6.1.11.5

Add $1$ and $1$ .

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

Step 1.6.1.11.6

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

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

Step 1.6.1.11.7

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

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

Step 1.6.1.11.8

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

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

Step 1.6.1.11.9

Add $1$ and $1$ .

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

Step 1.6.2

Add $4 \sin (2 x) \cos (2 x)$ and $4 \sin (2 x) \cos (2 x)$ .

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

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $16 \cos^{2} (x) \sin^{2} (x) - 8 \sin (2 x) \cos (2 x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 8 \sin (2 x) \cos (2 x) + 16 \sin^{2} (x) \cos^{2} (x)$ .

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Step 2.1.1

Add $- 8 \sin (2 x) \cos (2 x)$ and $8 \sin (2 x) \cos (2 x)$ .

$16 \cos^{2} (x) \sin^{2} (x) + 0 + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x) + 16 \sin^{2} (x) \cos^{2} (x)$

Step 2.1.2

Add $16 \cos^{2} (x) \sin^{2} (x)$ and $0$ .

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

Step 2.2

Reorder the factors of $16 \sin^{2} (x) \cos^{2} (x)$ .

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

Step 2.3

Add $16 \cos^{2} (x) \sin^{2} (x)$ and $16 \cos^{2} (x) \sin^{2} (x)$ .

$32 \cos^{2} (x) \sin^{2} (x) + 4 \cos^{2} (2 x) + 4 \cos^{2} (2 x)$

Step 2.4

Add $4 \cos^{2} (2 x)$ and $4 \cos^{2} (2 x)$ .

$32 \cos^{2} (x) \sin^{2} (x) + 8 \cos^{2} (2 x)$