Trigonometry Examples

$4 \cos (3 x) \cos (3 x)$

Step 1

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

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

Step 2

Apply the distributive property.

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

Step 3

Multiply.

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Step 3.1

Multiply $4$ by $4$ .

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

Step 3.2

Multiply $- 3$ by $4$ .

$(16 \cos^{3} (x) - 12 \cos (x)) \cos (3 x)$

Step 4

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

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

Step 5

Expand $(16 \cos^{3} (x) - 12 \cos (x)) (4 \cos^{3} (x) - 3 \cos (x))$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Rewrite using the commutative property of multiplication.

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

Step 6.1.2

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

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Step 6.1.2.1

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

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

Step 6.1.2.2

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

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

Step 6.1.2.3

Add $3$ and $3$ .

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

Step 6.1.3

Multiply $16$ by $4$ .

$64 \cos^{6} (x) + 16 \cos^{3} (x) (- 3 \cos (x)) - 12 \cos (x) (4 \cos^{3} (x)) - 12 \cos (x) (- 3 \cos (x))$

Step 6.1.4

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

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Step 6.1.4.1

Move $\cos (x)$ .

$64 \cos^{6} (x) + 16 (\cos (x) \cos^{3} (x)) \cdot - 3 - 12 \cos (x) (4 \cos^{3} (x)) - 12 \cos (x) (- 3 \cos (x))$

Step 6.1.4.2

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

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Step 6.1.4.2.1

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

$64 \cos^{6} (x) + 16 (\cos^{1} (x) \cos^{3} (x)) \cdot - 3 - 12 \cos (x) (4 \cos^{3} (x)) - 12 \cos (x) (- 3 \cos (x))$

Step 6.1.4.2.2

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

$64 \cos^{6} (x) + 16 {\cos (x)}^{1 + 3} \cdot - 3 - 12 \cos (x) (4 \cos^{3} (x)) - 12 \cos (x) (- 3 \cos (x))$

Step 6.1.4.3

Add $1$ and $3$ .

$64 \cos^{6} (x) + 16 \cos^{4} (x) \cdot - 3 - 12 \cos (x) (4 \cos^{3} (x)) - 12 \cos (x) (- 3 \cos (x))$

Step 6.1.5

Multiply $- 3$ by $16$ .

$64 \cos^{6} (x) - 48 \cos^{4} (x) - 12 \cos (x) (4 \cos^{3} (x)) - 12 \cos (x) (- 3 \cos (x))$

Step 6.1.6

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

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Step 6.1.6.1

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

$64 \cos^{6} (x) - 48 \cos^{4} (x) - 12 (\cos^{3} (x) \cos (x)) \cdot 4 - 12 \cos (x) (- 3 \cos (x))$

Step 6.1.6.2

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

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Step 6.1.6.2.1

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

$64 \cos^{6} (x) - 48 \cos^{4} (x) - 12 (\cos^{3} (x) \cos^{1} (x)) \cdot 4 - 12 \cos (x) (- 3 \cos (x))$

Step 6.1.6.2.2

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

$64 \cos^{6} (x) - 48 \cos^{4} (x) - 12 {\cos (x)}^{3 + 1} \cdot 4 - 12 \cos (x) (- 3 \cos (x))$

Step 6.1.6.3

Add $3$ and $1$ .

$64 \cos^{6} (x) - 48 \cos^{4} (x) - 12 \cos^{4} (x) \cdot 4 - 12 \cos (x) (- 3 \cos (x))$

Step 6.1.7

Multiply $4$ by $- 12$ .

$64 \cos^{6} (x) - 48 \cos^{4} (x) - 48 \cos^{4} (x) - 12 \cos (x) (- 3 \cos (x))$

Step 6.1.8

Multiply $- 12 \cos (x) (- 3 \cos (x))$ .

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Step 6.1.8.1

Multiply $- 3$ by $- 12$ .

$64 \cos^{6} (x) - 48 \cos^{4} (x) - 48 \cos^{4} (x) + 36 \cos (x) \cos (x)$

Step 6.1.8.2

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

$64 \cos^{6} (x) - 48 \cos^{4} (x) - 48 \cos^{4} (x) + 36 (\cos^{1} (x) \cos (x))$

Step 6.1.8.3

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

$64 \cos^{6} (x) - 48 \cos^{4} (x) - 48 \cos^{4} (x) + 36 (\cos^{1} (x) \cos^{1} (x))$

Step 6.1.8.4

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

$64 \cos^{6} (x) - 48 \cos^{4} (x) - 48 \cos^{4} (x) + 36 {\cos (x)}^{1 + 1}$

Step 6.1.8.5

Add $1$ and $1$ .

$64 \cos^{6} (x) - 48 \cos^{4} (x) - 48 \cos^{4} (x) + 36 \cos^{2} (x)$

Step 6.2

Subtract $48 \cos^{4} (x)$ from $- 48 \cos^{4} (x)$ .

$64 \cos^{6} (x) - 96 \cos^{4} (x) + 36 \cos^{2} (x)$