Trigonometry Examples

$\cos (6 x) + \cos (4 x)$

Step 1

Simplify each term.

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Step 1.1

Factor $2$ out of $6 x$ .

$\cos (2 (3 x)) + \cos (4 x)$

Step 1.2

Simplify each term.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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Step 1.2.3.1

Apply the distributive property.

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

Step 1.2.3.2

Apply the distributive property.

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

Step 1.2.3.3

Apply the distributive property.

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

Step 1.2.4

Simplify and combine like terms.

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Step 1.2.4.1

Simplify each term.

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Step 1.2.4.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.2.4.1.2

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

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Step 1.2.4.1.2.1

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

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

Step 1.2.4.1.2.2

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

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

Step 1.2.4.1.2.3

Add $3$ and $3$ .

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

Step 1.2.4.1.3

Multiply $4$ by $4$ .

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

Step 1.2.4.1.4

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

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Step 1.2.4.1.4.1

Move $\cos (x)$ .

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

Step 1.2.4.1.4.2

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

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Step 1.2.4.1.4.2.1

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

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

Step 1.2.4.1.4.2.2

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

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

Step 1.2.4.1.4.3

Add $1$ and $3$ .

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

Step 1.2.4.1.5

Multiply $- 3$ by $4$ .

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

Step 1.2.4.1.6

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

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Step 1.2.4.1.6.1

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

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

Step 1.2.4.1.6.2

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

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Step 1.2.4.1.6.2.1

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

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

Step 1.2.4.1.6.2.2

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

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

Step 1.2.4.1.6.3

Add $3$ and $1$ .

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

Step 1.2.4.1.7

Multiply $4$ by $- 3$ .

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

Step 1.2.4.1.8

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

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Step 1.2.4.1.8.1

Multiply $- 3$ by $- 3$ .

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

Step 1.2.4.1.8.2

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

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

Step 1.2.4.1.8.3

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

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

Step 1.2.4.1.8.4

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

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

Step 1.2.4.1.8.5

Add $1$ and $1$ .

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

Step 1.2.4.2

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

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

Step 1.2.5

Apply the distributive property.

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

Step 1.2.6

Simplify.

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Step 1.2.6.1

Multiply $16$ by $2$ .

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

Step 1.2.6.2

Multiply $- 24$ by $2$ .

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

Step 1.2.6.3

Multiply $9$ by $2$ .

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

Step 1.3

Factor $2$ out of $4 x$ .

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

Step 1.4

Simplify each term.

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Step 1.4.1

Use the double-angle identity to transform $\cos (2 x)$ to $2 \cos^{2} (x) - 1$ .

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

Step 1.4.2

Rewrite ${(2 \cos^{2} (x) - 1)}^{2}$ as $(2 \cos^{2} (x) - 1) (2 \cos^{2} (x) - 1)$ .

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

Step 1.4.3

Expand $(2 \cos^{2} (x) - 1) (2 \cos^{2} (x) - 1)$ using the FOIL Method.

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Step 1.4.3.1

Apply the distributive property.

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

Step 1.4.3.2

Apply the distributive property.

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

Step 1.4.3.3

Apply the distributive property.

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

Step 1.4.4

Simplify and combine like terms.

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Step 1.4.4.1

Simplify each term.

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Step 1.4.4.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.4.4.1.2

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

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Step 1.4.4.1.2.1

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

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

Step 1.4.4.1.2.2

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

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

Step 1.4.4.1.2.3

Add $2$ and $2$ .

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

Step 1.4.4.1.3

Multiply $2$ by $2$ .

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

Step 1.4.4.1.4

Multiply $- 1$ by $2$ .

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

Step 1.4.4.1.5

Multiply $2$ by $- 1$ .

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

Step 1.4.4.1.6

Multiply $- 1$ by $- 1$ .

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

Step 1.4.4.2

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

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

Step 1.4.5

Apply the distributive property.

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

Step 1.4.6

Simplify.

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Step 1.4.6.1

Multiply $4$ by $2$ .

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

Step 1.4.6.2

Multiply $- 4$ by $2$ .

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

Step 1.4.6.3

Multiply $2$ by $1$ .

$32 \cos^{6} (x) - 48 \cos^{4} (x) + 18 \cos^{2} (x) - 1 + 8 \cos^{4} (x) - 8 \cos^{2} (x) + 2 - 1$

Step 1.5

Subtract $1$ from $2$ .

$32 \cos^{6} (x) - 48 \cos^{4} (x) + 18 \cos^{2} (x) - 1 + 8 \cos^{4} (x) - 8 \cos^{2} (x) + 1$

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $32 \cos^{6} (x) - 48 \cos^{4} (x) + 18 \cos^{2} (x) - 1 + 8 \cos^{4} (x) - 8 \cos^{2} (x) + 1$ .

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Step 2.1.1

Add $- 1$ and $1$ .

$32 \cos^{6} (x) - 48 \cos^{4} (x) + 18 \cos^{2} (x) + 0 + 8 \cos^{4} (x) - 8 \cos^{2} (x)$

Step 2.1.2

Add $32 \cos^{6} (x) - 48 \cos^{4} (x) + 18 \cos^{2} (x)$ and $0$ .

$32 \cos^{6} (x) - 48 \cos^{4} (x) + 18 \cos^{2} (x) + 8 \cos^{4} (x) - 8 \cos^{2} (x)$

Step 2.2

Add $- 48 \cos^{4} (x)$ and $8 \cos^{4} (x)$ .

$32 \cos^{6} (x) - 40 \cos^{4} (x) + 18 \cos^{2} (x) - 8 \cos^{2} (x)$

Step 2.3

Subtract $8 \cos^{2} (x)$ from $18 \cos^{2} (x)$ .

$32 \cos^{6} (x) - 40 \cos^{4} (x) + 10 \cos^{2} (x)$