Trigonometry Examples

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

Step 1

Simplify each term.

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Step 1.1

Factor $2$ out of $4 x$ .

$\cos (2 (2 x)) - \cos (2 x)$

Step 1.2

Simplify each term.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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Step 1.2.3.1

Apply the distributive property.

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

Step 1.2.3.2

Apply the distributive property.

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

Step 1.2.3.3

Apply the distributive property.

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

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.

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

Step 1.2.4.1.2

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

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Step 1.2.4.1.2.1

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

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

Step 1.2.4.1.2.2

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

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

Step 1.2.4.1.2.3

Add $2$ and $2$ .

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

Step 1.2.4.1.3

Multiply $2$ by $2$ .

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

Step 1.2.4.1.4

Multiply $- 1$ by $2$ .

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

Step 1.2.4.1.5

Multiply $2$ by $- 1$ .

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

Step 1.2.4.1.6

Multiply $- 1$ by $- 1$ .

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

Step 1.2.4.2

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

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

Step 1.2.5

Apply the distributive property.

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

Step 1.2.6

Simplify.

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Step 1.2.6.1

Multiply $4$ by $2$ .

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

Step 1.2.6.2

Multiply $- 4$ by $2$ .

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

Step 1.2.6.3

Multiply $2$ by $1$ .

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

Step 1.3

Subtract $1$ from $2$ .

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

Step 1.4

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

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

Step 1.5

Apply the distributive property.

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

Step 1.6

Multiply $2$ by $- 1$ .

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

Step 1.7

Multiply $- 1$ by $- 1$ .

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

Step 2

Simplify by adding terms.

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Step 2.1

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

$8 \cos^{4} (x) + 1 - 10 \cos^{2} (x) + 1$

Step 2.2

Add $1$ and $1$ .

$8 \cos^{4} (x) + 2 - 10 \cos^{2} (x)$