Trigonometry Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

Multiply $4$ by $4$ .

$16 \sin (x) \sin (x) + 4 \sin (x) (- 4 \cos (x)) - 4 \cos (x) (4 \sin (x)) - 4 \cos (x) (- 4 \cos (x))$

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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

Step 3.1.1.4

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

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

Step 3.1.1.5

Add $1$ and $1$ .

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

Step 3.1.2

Multiply $- 4$ by $4$ .

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

Step 3.1.3

Multiply $4$ by $- 4$ .

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

Step 3.1.4

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

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Step 3.1.4.1

Multiply $- 4$ by $- 4$ .

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

Step 3.1.4.2

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

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

Step 3.1.4.3

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

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

Step 3.1.4.4

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

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

Step 3.1.4.5

Add $1$ and $1$ .

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

Step 3.2

Reorder the factors of $- 16 \sin (x) \cos (x)$ .

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

Step 3.3

Subtract $16 \cos (x) \sin (x)$ from $- 16 \cos (x) \sin (x)$ .

$16 \sin^{2} (x) - 32 \cos (x) \sin (x) + 16 \cos^{2} (x)$

Step 4

Simplify with factoring out.

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Step 4.1

Move $16 \cos^{2} (x)$ .

$16 \sin^{2} (x) + 16 \cos^{2} (x) - 32 \cos (x) \sin (x)$

Step 4.2

Factor $16$ out of $16 \sin^{2} (x)$ .

$16 (\sin^{2} (x)) + 16 \cos^{2} (x) - 32 \cos (x) \sin (x)$

Step 4.3

Factor $16$ out of $16 \cos^{2} (x)$ .

$16 (\sin^{2} (x)) + 16 (\cos^{2} (x)) - 32 \cos (x) \sin (x)$

Step 4.4

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

$16 (\sin^{2} (x) + \cos^{2} (x)) - 32 \cos (x) \sin (x)$

Step 5

Apply pythagorean identity.

$16 \cdot 1 - 32 \cos (x) \sin (x)$

Step 6

Multiply $16$ by $1$ .

$16 - 32 \cos (x) \sin (x)$