Trigonometry Examples

$\cos (2 x) - \cos (6 x) = 0$

Step 1

Simplify the left side of the equation.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$2 \cos^{2} (x) - 1 - \cos (6 x) = 0$

Step 1.1.2

Factor $2$ out of $6 x$ .

$2 \cos^{2} (x) - 1 - \cos (2 (3 x)) = 0$

Step 1.1.3

Simplify each term.

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Step 1.1.3.1

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

$2 \cos^{2} (x) - 1 - (2 {(4 \cos^{3} (x) - 3 \cos (x))}^{2} - 1) = 0$

Step 1.1.3.2

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

$2 \cos^{2} (x) - 1 - (2 ((4 \cos^{3} (x) - 3 \cos (x)) (4 \cos^{3} (x) - 3 \cos (x))) - 1) = 0$

Step 1.1.3.3

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

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Step 1.1.3.3.1

Apply the distributive property.

$2 \cos^{2} (x) - 1 - (2 (4 \cos^{3} (x) (4 \cos^{3} (x) - 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x) - 3 \cos (x))) - 1) = 0$

Step 1.1.3.3.2

Apply the distributive property.

$2 \cos^{2} (x) - 1 - (2 (4 \cos^{3} (x) (4 \cos^{3} (x)) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x) - 3 \cos (x))) - 1) = 0$

Step 1.1.3.3.3

Apply the distributive property.

$2 \cos^{2} (x) - 1 - (2 (4 \cos^{3} (x) (4 \cos^{3} (x)) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x))) - 1) = 0$

Step 1.1.3.4

Simplify and combine like terms.

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Step 1.1.3.4.1

Simplify each term.

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Step 1.1.3.4.1.1

Rewrite using the commutative property of multiplication.

$2 \cos^{2} (x) - 1 - (2 (4 \cdot (4 \cos^{3} (x) \cos^{3} (x)) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x))) - 1) = 0$

Step 1.1.3.4.1.2

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

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Step 1.1.3.4.1.2.1

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

$2 \cos^{2} (x) - 1 - (2 (4 \cdot (4 (\cos^{3} (x) \cos^{3} (x))) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x))) - 1) = 0$

Step 1.1.3.4.1.2.2

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

$2 \cos^{2} (x) - 1 - (2 (4 \cdot (4 {\cos (x)}^{3 + 3}) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x))) - 1) = 0$

Step 1.1.3.4.1.2.3

Add $3$ and $3$ .

$2 \cos^{2} (x) - 1 - (2 (4 \cdot (4 \cos^{6} (x)) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x))) - 1) = 0$

Step 1.1.3.4.1.3

Multiply $4$ by $4$ .

$2 \cos^{2} (x) - 1 - (2 (16 \cos^{6} (x) + 4 \cos^{3} (x) (- 3 \cos (x)) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x))) - 1) = 0$

Step 1.1.3.4.1.4

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

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Step 1.1.3.4.1.4.1

Move $\cos (x)$ .

$2 \cos^{2} (x) - 1 - (2 (16 \cos^{6} (x) + 4 (\cos (x) \cos^{3} (x)) \cdot - 3 - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x))) - 1) = 0$

Step 1.1.3.4.1.4.2

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

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Step 1.1.3.4.1.4.2.1

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

$2 \cos^{2} (x) - 1 - (2 (16 \cos^{6} (x) + 4 (\cos (x) \cos^{3} (x)) \cdot - 3 - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x))) - 1) = 0$

Step 1.1.3.4.1.4.2.2

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

$2 \cos^{2} (x) - 1 - (2 (16 \cos^{6} (x) + 4 {\cos (x)}^{1 + 3} \cdot - 3 - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x))) - 1) = 0$

Step 1.1.3.4.1.4.3

Add $1$ and $3$ .

$2 \cos^{2} (x) - 1 - (2 (16 \cos^{6} (x) + 4 \cos^{4} (x) \cdot - 3 - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x))) - 1) = 0$

Step 1.1.3.4.1.5

Multiply $- 3$ by $4$ .

$2 \cos^{2} (x) - 1 - (2 (16 \cos^{6} (x) - 12 \cos^{4} (x) - 3 \cos (x) (4 \cos^{3} (x)) - 3 \cos (x) (- 3 \cos (x))) - 1) = 0$

Step 1.1.3.4.1.6

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

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Step 1.1.3.4.1.6.1

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

$2 \cos^{2} (x) - 1 - (2 (16 \cos^{6} (x) - 12 \cos^{4} (x) - 3 (\cos^{3} (x) \cos (x)) \cdot 4 - 3 \cos (x) (- 3 \cos (x))) - 1) = 0$

Step 1.1.3.4.1.6.2

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

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Step 1.1.3.4.1.6.2.1

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

$2 \cos^{2} (x) - 1 - (2 (16 \cos^{6} (x) - 12 \cos^{4} (x) - 3 (\cos^{3} (x) \cos (x)) \cdot 4 - 3 \cos (x) (- 3 \cos (x))) - 1) = 0$

Step 1.1.3.4.1.6.2.2

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

$2 \cos^{2} (x) - 1 - (2 (16 \cos^{6} (x) - 12 \cos^{4} (x) - 3 {\cos (x)}^{3 + 1} \cdot 4 - 3 \cos (x) (- 3 \cos (x))) - 1) = 0$

Step 1.1.3.4.1.6.3

Add $3$ and $1$ .

$2 \cos^{2} (x) - 1 - (2 (16 \cos^{6} (x) - 12 \cos^{4} (x) - 3 \cos^{4} (x) \cdot 4 - 3 \cos (x) (- 3 \cos (x))) - 1) = 0$

Step 1.1.3.4.1.7

Multiply $4$ by $- 3$ .

$2 \cos^{2} (x) - 1 - (2 (16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) - 3 \cos (x) (- 3 \cos (x))) - 1) = 0$

Step 1.1.3.4.1.8

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

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Step 1.1.3.4.1.8.1

Multiply $- 3$ by $- 3$ .

$2 \cos^{2} (x) - 1 - (2 (16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos (x) \cos (x)) - 1) = 0$

Step 1.1.3.4.1.8.2

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

$2 \cos^{2} (x) - 1 - (2 (16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 (\cos (x) \cos (x))) - 1) = 0$

Step 1.1.3.4.1.8.3

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

$2 \cos^{2} (x) - 1 - (2 (16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 (\cos (x) \cos (x))) - 1) = 0$

Step 1.1.3.4.1.8.4

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

$2 \cos^{2} (x) - 1 - (2 (16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 {\cos (x)}^{1 + 1}) - 1) = 0$

Step 1.1.3.4.1.8.5

Add $1$ and $1$ .

$2 \cos^{2} (x) - 1 - (2 (16 \cos^{6} (x) - 12 \cos^{4} (x) - 12 \cos^{4} (x) + 9 \cos^{2} (x)) - 1) = 0$

Step 1.1.3.4.2

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

$2 \cos^{2} (x) - 1 - (2 (16 \cos^{6} (x) - 24 \cos^{4} (x) + 9 \cos^{2} (x)) - 1) = 0$

Step 1.1.3.5

Apply the distributive property.

$2 \cos^{2} (x) - 1 - (2 (16 \cos^{6} (x)) + 2 (- 24 \cos^{4} (x)) + 2 (9 \cos^{2} (x)) - 1) = 0$

Step 1.1.3.6

Simplify.

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Step 1.1.3.6.1

Multiply $16$ by $2$ .

$2 \cos^{2} (x) - 1 - (32 \cos^{6} (x) + 2 (- 24 \cos^{4} (x)) + 2 (9 \cos^{2} (x)) - 1) = 0$

Step 1.1.3.6.2

Multiply $- 24$ by $2$ .

$2 \cos^{2} (x) - 1 - (32 \cos^{6} (x) - 48 \cos^{4} (x) + 2 (9 \cos^{2} (x)) - 1) = 0$

Step 1.1.3.6.3

Multiply $9$ by $2$ .

$2 \cos^{2} (x) - 1 - (32 \cos^{6} (x) - 48 \cos^{4} (x) + 18 \cos^{2} (x) - 1) = 0$

Step 1.1.4

Apply the distributive property.

$2 \cos^{2} (x) - 1 - (32 \cos^{6} (x)) - (- 48 \cos^{4} (x)) - (18 \cos^{2} (x)) + 1 = 0$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Multiply $32$ by $- 1$ .

$2 \cos^{2} (x) - 1 - 32 \cos^{6} (x) - (- 48 \cos^{4} (x)) - (18 \cos^{2} (x)) + 1 = 0$

Step 1.1.5.2

Multiply $- 48$ by $- 1$ .

$2 \cos^{2} (x) - 1 - 32 \cos^{6} (x) + 48 \cos^{4} (x) - (18 \cos^{2} (x)) + 1 = 0$

Step 1.1.5.3

Multiply $18$ by $- 1$ .

$2 \cos^{2} (x) - 1 - 32 \cos^{6} (x) + 48 \cos^{4} (x) - 18 \cos^{2} (x) + 1 = 0$

Step 1.1.5.4

Multiply $- 1$ by $- 1$ .

$2 \cos^{2} (x) - 1 - 32 \cos^{6} (x) + 48 \cos^{4} (x) - 18 \cos^{2} (x) + 1 = 0$

Step 1.2

Simplify by adding terms.

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Step 1.2.1

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

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Step 1.2.1.1

Add $- 1$ and $1$ .

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

Step 1.2.1.2

Add $2 \cos^{2} (x)$ and $0$ .

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

Step 1.2.2

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

$- 16 \cos^{2} (x) - 32 \cos^{6} (x) + 48 \cos^{4} (x) = 0$

Step 2

Factor $- 16 \cos^{2} (x) - 32 \cos^{6} (x) + 48 \cos^{4} (x)$ .

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Step 2.1

Factor $16 \cos^{2} (x)$ out of $- 16 \cos^{2} (x) - 32 \cos^{6} (x) + 48 \cos^{4} (x)$ .

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Step 2.1.1

Factor $16 \cos^{2} (x)$ out of $- 16 \cos^{2} (x)$ .

$16 \cos^{2} (x) \cdot - 1 - 32 \cos^{6} (x) + 48 \cos^{4} (x) = 0$

Step 2.1.2

Factor $16 \cos^{2} (x)$ out of $- 32 \cos^{6} (x)$ .

$16 \cos^{2} (x) \cdot - 1 + 16 \cos^{2} (x) (- 2 \cos^{4} (x)) + 48 \cos^{4} (x) = 0$

Step 2.1.3

Factor $16 \cos^{2} (x)$ out of $48 \cos^{4} (x)$ .

$16 \cos^{2} (x) \cdot - 1 + 16 \cos^{2} (x) (- 2 \cos^{4} (x)) + 16 \cos^{2} (x) (3 \cos^{2} (x)) = 0$

Step 2.1.4

Factor $16 \cos^{2} (x)$ out of $16 \cos^{2} (x) \cdot - 1 + 16 \cos^{2} (x) (- 2 \cos^{4} (x))$ .

$16 \cos^{2} (x) (- 1 - 2 \cos^{4} (x)) + 16 \cos^{2} (x) (3 \cos^{2} (x)) = 0$

Step 2.1.5

Factor $16 \cos^{2} (x)$ out of $16 \cos^{2} (x) (- 1 - 2 \cos^{4} (x)) + 16 \cos^{2} (x) (3 \cos^{2} (x))$ .

$16 \cos^{2} (x) (- 1 - 2 \cos^{4} (x) + 3 \cos^{2} (x)) = 0$

Step 2.2

Factor by grouping.

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Step 2.2.1

Reorder terms.

$16 \cos^{2} (x) (- 2 \cos^{4} (x) + 3 \cos^{2} (x) - 1) = 0$

Step 2.2.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 2 \cdot - 1 = 2$ and whose sum is $b = 3$ .

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Step 2.2.2.1

Factor $3$ out of $3 \cos^{2} (x)$ .

$16 \cos^{2} (x) (- 2 \cos^{4} (x) + 3 \cos^{2} (x) - 1) = 0$

Step 2.2.2.2

Rewrite $3$ as $1$ plus $2$

$16 \cos^{2} (x) (- 2 \cos^{4} (x) + (1 + 2) \cos^{2} (x) - 1) = 0$

Step 2.2.2.3

Apply the distributive property.

$16 \cos^{2} (x) (- 2 \cos^{4} (x) + 1 \cos^{2} (x) + 2 \cos^{2} (x) - 1) = 0$

Step 2.2.2.4

Multiply $\cos^{2} (x)$ by $1$ .

$16 \cos^{2} (x) (- 2 \cos^{4} (x) + \cos^{2} (x) + 2 \cos^{2} (x) - 1) = 0$

Step 2.2.3

Factor out the greatest common factor from each group.

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Step 2.2.3.1

Group the first two terms and the last two terms.

$16 \cos^{2} (x) (- 2 \cos^{4} (x) + \cos^{2} (x) + 2 \cos^{2} (x) - 1) = 0$

Step 2.2.3.2

Factor out the greatest common factor (GCF) from each group.

$16 \cos^{2} (x) (\cos^{2} (x) (- 2 \cos^{2} (x) + 1) - (- 2 \cos^{2} (x) + 1)) = 0$

Step 2.2.4

Factor the polynomial by factoring out the greatest common factor, $- 2 \cos^{2} (x) + 1$ .

$16 \cos^{2} (x) ((- 2 \cos^{2} (x) + 1) (\cos^{2} (x) - 1)) = 0$

Step 2.3

Rewrite $1$ as $1^{2}$ .

$16 \cos^{2} (x) ((- 2 \cos^{2} (x) + 1) (\cos^{2} (x) - 1^{2})) = 0$

Step 2.4

Factor.

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Step 2.4.1

Factor.

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Step 2.4.1.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \cos (x)$ and $b = 1$ .

$16 \cos^{2} (x) ((- 2 \cos^{2} (x) + 1) ((\cos (x) + 1) (\cos (x) - 1))) = 0$

Step 2.4.1.2

Remove unnecessary parentheses.

$16 \cos^{2} (x) ((- 2 \cos^{2} (x) + 1) (\cos (x) + 1) (\cos (x) - 1)) = 0$

Step 2.4.2

Remove unnecessary parentheses.

$16 \cos^{2} (x) (- 2 \cos^{2} (x) + 1) (\cos (x) + 1) (\cos (x) - 1) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\cos^{2} (x) = 0$

$- 2 \cos^{2} (x) + 1 = 0$

$\cos (x) + 1 = 0$

$\cos (x) - 1 = 0$

Step 4

Set $\cos^{2} (x)$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $\cos^{2} (x)$ equal to $0$ .

$\cos^{2} (x) = 0$

Step 4.2

Solve $\cos^{2} (x) = 0$ for $x$ .

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Step 4.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\cos (x) = \pm \sqrt{0}$

Step 4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 4.2.2.1

Rewrite $0$ as $0^{2}$ .

$\cos (x) = \pm \sqrt{0^{2}}$

Step 4.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$\cos (x) = \pm 0$

Step 4.2.2.3

Plus or minus $0$ is $0$ .

$\cos (x) = 0$

Step 4.2.3

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (0)$

Step 4.2.4

Simplify the right side.

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Step 4.2.4.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 4.2.5

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{2}$

Step 4.2.6

Simplify $2 π - \frac{π}{2}$ .

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Step 4.2.6.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 4.2.6.2

Combine fractions.

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Step 4.2.6.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$x = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 4.2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 4.2.6.3

Simplify the numerator.

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Step 4.2.6.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 4.2.6.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 4.2.7

Find the period of $\cos (x)$ .

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Step 4.2.7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 4.2.7.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 4.2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 4.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.2.8

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 5

Set $- 2 \cos^{2} (x) + 1$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $- 2 \cos^{2} (x) + 1$ equal to $0$ .

$- 2 \cos^{2} (x) + 1 = 0$

Step 5.2

Solve $- 2 \cos^{2} (x) + 1 = 0$ for $x$ .

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Step 5.2.1

Subtract $1$ from both sides of the equation.

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

Step 5.2.2

Divide each term in $- 2 \cos^{2} (x) = - 1$ by $- 2$ and simplify.

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Step 5.2.2.1

Divide each term in $- 2 \cos^{2} (x) = - 1$ by $- 2$ .

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

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.2.1.2

Divide $\cos^{2} (x)$ by $1$ .

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

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Dividing two negative values results in a positive value.

$\cos^{2} (x) = \frac{1}{2}$

Step 5.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\cos (x) = \pm \sqrt{\frac{1}{2}}$

Step 5.2.4

Simplify $\pm \sqrt{\frac{1}{2}}$ .

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Step 5.2.4.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$\cos (x) = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Step 5.2.4.2

Any root of $1$ is $1$ .

$\cos (x) = \pm \frac{1}{\sqrt{2}}$

Step 5.2.4.3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cos (x) = \pm \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 5.2.4.4

Combine and simplify the denominator.

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Step 5.2.4.4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cos (x) = \pm \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.2.4.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$\cos (x) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 5.2.4.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$\cos (x) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 5.2.4.4.4

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

$\cos (x) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 5.2.4.4.5

Add $1$ and $1$ .

$\cos (x) = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 5.2.4.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 5.2.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\cos (x) = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 5.2.4.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\cos (x) = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 5.2.4.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$\cos (x) = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.2.4.4.6.4

Cancel the common factor of $2$ .

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Step 5.2.4.4.6.4.1

Cancel the common factor.

$\cos (x) = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.2.4.4.6.4.2

Rewrite the expression.

$\cos (x) = \pm \frac{\sqrt{2}}{2^{1}}$

Step 5.2.4.4.6.5

Evaluate the exponent.

$\cos (x) = \pm \frac{\sqrt{2}}{2}$

Step 5.2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$\cos (x) = \frac{\sqrt{2}}{2}$

Step 5.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$\cos (x) = - \frac{\sqrt{2}}{2}$

Step 5.2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$\cos (x) = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Step 5.2.6

Set up each of the solutions to solve for $x$ .

$\cos (x) = \frac{\sqrt{2}}{2}$

$\cos (x) = - \frac{\sqrt{2}}{2}$

Step 5.2.7

Solve for $x$ in $\cos (x) = \frac{\sqrt{2}}{2}$ .

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Step 5.2.7.1

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (\frac{\sqrt{2}}{2})$

Step 5.2.7.2

Simplify the right side.

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Step 5.2.7.2.1

The exact value of $\arccos (\frac{\sqrt{2}}{2})$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 5.2.7.3

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{4}$

Step 5.2.7.4

Simplify $2 π - \frac{π}{4}$ .

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Step 5.2.7.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{π}{4}$

Step 5.2.7.4.2

Combine fractions.

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Step 5.2.7.4.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{π}{4}$

Step 5.2.7.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - π}{4}$

Step 5.2.7.4.3

Simplify the numerator.

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Step 5.2.7.4.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - π}{4}$

Step 5.2.7.4.3.2

Subtract $π$ from $8 π$ .

$x = \frac{7 π}{4}$

Step 5.2.7.5

Find the period of $\cos (x)$ .

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Step 5.2.7.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 5.2.7.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 5.2.7.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 5.2.7.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.7.6

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 5.2.8

Solve for $x$ in $\cos (x) = - \frac{\sqrt{2}}{2}$ .

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Step 5.2.8.1

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (- \frac{\sqrt{2}}{2})$

Step 5.2.8.2

Simplify the right side.

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Step 5.2.8.2.1

The exact value of $\arccos (- \frac{\sqrt{2}}{2})$ is $\frac{3 π}{4}$ .

$x = \frac{3 π}{4}$

Step 5.2.8.3

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 π - \frac{3 π}{4}$

Step 5.2.8.4

Simplify $2 π - \frac{3 π}{4}$ .

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Step 5.2.8.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{3 π}{4}$

Step 5.2.8.4.2

Combine fractions.

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Step 5.2.8.4.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{3 π}{4}$

Step 5.2.8.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - 3 π}{4}$

Step 5.2.8.4.3

Simplify the numerator.

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Step 5.2.8.4.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - 3 π}{4}$

Step 5.2.8.4.3.2

Subtract $3 π$ from $8 π$ .

$x = \frac{5 π}{4}$

Step 5.2.8.5

Find the period of $\cos (x)$ .

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Step 5.2.8.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 5.2.8.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 5.2.8.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 5.2.8.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.8.6

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n$ , for any integer $n$

Step 5.2.9

List all of the solutions.

$x = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n, \frac{3 π}{4} + 2 π n, \frac{5 π}{4} + 2 π n$ , for any integer $n$

Step 5.2.10

Consolidate the answers.

$x = \frac{π}{4} + \frac{π n}{2}$ , for any integer $n$

Step 6

Set $\cos (x) + 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $\cos (x) + 1$ equal to $0$ .

$\cos (x) + 1 = 0$

Step 6.2

Solve $\cos (x) + 1 = 0$ for $x$ .

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Step 6.2.1

Subtract $1$ from both sides of the equation.

$\cos (x) = - 1$

Step 6.2.2

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (- 1)$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

The exact value of $\arccos (- 1)$ is $π$ .

$x = π$

Step 6.2.4

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 π - π$

Step 6.2.5

Subtract $π$ from $2 π$ .

$x = π$

Step 6.2.6

Find the period of $\cos (x)$ .

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Step 6.2.6.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 6.2.6.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 6.2.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 6.2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.2.7

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = π + 2 π n$ , for any integer $n$

Step 7

Set $\cos (x) - 1$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $\cos (x) - 1$ equal to $0$ .

$\cos (x) - 1 = 0$

Step 7.2

Solve $\cos (x) - 1 = 0$ for $x$ .

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Step 7.2.1

Add $1$ to both sides of the equation.

$\cos (x) = 1$

Step 7.2.2

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$x = \arccos (1)$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

The exact value of $\arccos (1)$ is $0$ .

$x = 0$

Step 7.2.4

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - 0$

Step 7.2.5

Subtract $0$ from $2 π$ .

$x = 2 π$

Step 7.2.6

Find the period of $\cos (x)$ .

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Step 7.2.6.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 7.2.6.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 7.2.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 7.2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 7.2.7

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 2 π n, 2 π + 2 π n$ , for any integer $n$

Step 8

The final solution is all the values that make $16 \cos^{2} (x) (- 2 \cos^{2} (x) + 1) (\cos (x) + 1) (\cos (x) - 1) = 0$ true.

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n, \frac{π}{4} + \frac{π n}{2}, π + 2 π n, 2 π n, 2 π + 2 π n$ , for any integer $n$

Step 9

Consolidate the answers.

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Step 9.1

Consolidate $\frac{π}{2} + 2 π n$ and $\frac{3 π}{2} + 2 π n$ to $\frac{π}{2} + π n$ .

$x = \frac{π}{2} + π n, \frac{π}{4} + \frac{π n}{2}, π + 2 π n, 2 π n, 2 π + 2 π n$ , for any integer $n$

Step 9.2

Consolidate $π + 2 π n$ and $2 π n$ to $π n$ .

$x = \frac{π}{2} + π n, \frac{π}{4} + \frac{π n}{2}, π n, 2 π + 2 π n$ , for any integer $n$

Step 9.3

Consolidate $\frac{π}{2} + π n$ and $π n$ to $\frac{π n}{2}$ .

$x = \frac{π n}{2}, \frac{π}{4} + \frac{π n}{2}, 2 π + 2 π n$ , for any integer $n$

Step 9.4

Consolidate $\frac{π n}{2}$ and $\frac{π}{4} + \frac{π n}{2}$ to $\frac{π n}{4}$ .

$x = \frac{π n}{4}, 2 π + 2 π n$ , for any integer $n$

Step 9.5

Consolidate $\frac{π n}{4}$ and $2 π + 2 π n$ to $\frac{π n}{4}$ .

$x = \frac{π n}{4}$ , for any integer $n$