Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

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

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Step 1.3.1.1.1

Multiply $2$ by $2$ .

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

Step 1.3.1.1.2

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

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

Step 1.3.1.1.3

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

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

Step 1.3.1.1.4

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

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

Step 1.3.1.1.5

Add $1$ and $1$ .

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

Step 1.3.1.2

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

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

Step 1.3.1.3

Add parentheses.

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

Step 1.3.1.4

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

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

Step 1.3.1.5

Apply the sine double-angle identity.

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

Step 1.3.1.6

Add parentheses.

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

Step 1.3.1.7

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

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

Step 1.3.1.8

Apply the sine double-angle identity.

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

Step 1.3.1.9

Multiply $- 2 \sin (x) (- 2 \sin (x))$ .

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Step 1.3.1.9.1

Multiply $- 2$ by $- 2$ .

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

Step 1.3.1.9.2

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

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

Step 1.3.1.9.3

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

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

Step 1.3.1.9.4

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

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

Step 1.3.1.9.5

Add $1$ and $1$ .

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

Step 1.3.2

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

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

Step 1.4

Move $4 \sin^{2} (x)$ .

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

Step 1.5

Factor $4$ out of $4 \cos^{2} (x)$ .

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

Step 1.6

Factor $4$ out of $4 \sin^{2} (x)$ .

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

Step 1.7

Factor $4$ out of $4 (\cos^{2} (x)) + 4 (\sin^{2} (x))$ .

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

Step 1.8

Rearrange terms.

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

Step 1.9

Apply pythagorean identity.

$4 \cdot 1 - 4 \sin (2 x) + 4 \sin (2 x)$

Step 1.10

Multiply $4$ by $1$ .

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

Step 2

Combine the opposite terms in $4 - 4 \sin (2 x) + 4 \sin (2 x)$ .

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Step 2.1

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

$4 + 0$

Step 2.2

Add $4$ and $0$ .

$4$