Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite using the commutative property of multiplication.

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

Step 4.1.2

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

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Step 4.1.2.1

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

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

Step 4.1.2.2

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

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

Step 4.1.2.3

Add $2$ and $2$ .

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

Step 4.1.3

Multiply $2$ by $2$ .

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

Step 4.1.4

Multiply $- 1$ by $2$ .

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

Step 4.1.5

Multiply $2$ by $- 1$ .

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

Step 4.1.6

Multiply $- 1$ by $- 1$ .

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

Step 4.2

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

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