Simplify (2cos(theta)-1)(2cos(theta)+1)

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Algebra

$(2 \cos (θ) - 1) (2 \cos (θ) + 1)$

Expand $(2 \cos (θ) - 1) (2 \cos (θ) + 1)$ using the FOIL Method.

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Apply the distributive property.

$2 \cos (θ) (2 \cos (θ) + 1) - 1 (2 \cos (θ) + 1)$

Apply the distributive property.

$2 \cos (θ) (2 \cos (θ)) + 2 \cos (θ) \cdot 1 - 1 (2 \cos (θ) + 1)$

Apply the distributive property.

$2 \cos (θ) (2 \cos (θ)) + 2 \cos (θ) \cdot 1 - 1 (2 \cos (θ)) - 1 \cdot 1$

Simplify terms.

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Combine the opposite terms in $2 \cos (θ) (2 \cos (θ)) + 2 \cos (θ) \cdot 1 - 1 (2 \cos (θ)) - 1 \cdot 1$ .

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Reorder the factors in the terms $2 \cos (θ) \cdot 1$ and $- 1 (2 \cos (θ))$ .

$2 \cos (θ) (2 \cos (θ)) + 1 \cdot 2 \cos (θ) - 1 \cdot 2 \cos (θ) - 1 \cdot 1$

Subtract $2 \cos (θ)$ from $1 \cdot 2 \cos (θ)$ .

$2 \cos (θ) (2 \cos (θ)) + 0 - 1 \cdot 1$

Add $2 \cos (θ) (2 \cos (θ))$ and $0$ .

$2 \cos (θ) (2 \cos (θ)) - 1 \cdot 1$

Simplify each term.

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Multiply $2 \cos (θ) (2 \cos (θ))$ .

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Multiply $2$ by $2$ .

$4 \cos (θ) \cos (θ) - 1 \cdot 1$

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

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

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

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

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

$4 {\cos (θ)}^{1 + 1} - 1 \cdot 1$

Add $1$ and $1$ .

$4 \cos^{2} (θ) - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$4 \cos^{2} (θ) - 1$

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