Algebra Examples

$\cos (2 u) = \cos^{2} (u) - \sin^{2} (u)$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\cos^{2} (u)$ from both sides of the equation.

$\cos (2 u) - \cos^{2} (u) = - \sin^{2} (u)$

Step 1.2

Add $\sin^{2} (u)$ to both sides of the equation.

$\cos (2 u) - \cos^{2} (u) + \sin^{2} (u) = 0$

Step 2

Replace $\sin^{2} (u)$ with $1 - \cos^{2} (u)$ .

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

Step 3

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

$1 - 2 \sin^{2} (u) - \cos^{2} (u) (1 - \cos^{2} (u)) = 0$

Step 4

Subtract $1$ from both sides of the equation.

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

Step 5

Simplify the left side.

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Step 5.1

Apply pythagorean identity.

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

Step 6

Solve the equation for $u$ .

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Step 6.1

Add $1$ to both sides of the equation.

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

Step 6.2

Replace the $- 2 \sin^{2} (u)$ with $- 2 (1 - \cos^{2} (u))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

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

Step 6.3

Simplify each term.

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Step 6.3.1

Apply the distributive property.

$- 2 \cdot 1 - 2 (- \cos^{2} (u)) - \cos^{2} (u) \sin^{2} (u) + 1 = 0$

Step 6.3.2

Multiply $- 2$ by $1$ .

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

Step 6.3.3

Multiply $- 1$ by $- 2$ .

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

Step 6.4

Add $- 2$ and $1$ .

$2 \cos^{2} (u) - \cos^{2} (u) \sin^{2} (u) - 1 = 0$

Step 6.5

Simplify the left side.

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Step 6.5.1

Simplify $2 \cos^{2} (u) - \cos^{2} (u) \sin^{2} (u) - 1$ .

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Step 6.5.1.1

Move $- 1$ .

$2 \cos^{2} (u) - 1 - \cos^{2} (u) \sin^{2} (u) = 0$

Step 6.5.1.2

Apply the cosine double-angle identity.

$\cos (2 u) - \cos^{2} (u) \sin^{2} (u) = 0$

Step 6.6

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

$1 - 2 \sin^{2} (u) - \cos^{2} (u) \sin^{2} (u) = 0$

Step 6.7

Subtract $1$ from both sides of the equation.

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

Step 6.8

Solve the equation for $u$ .

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Step 6.8.1

Add $1$ to both sides of the equation.

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

Step 6.8.2

Replace the $- 2 \sin^{2} (u)$ with $- 2 (1 - \cos^{2} (u))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

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

Step 6.8.3

Simplify each term.

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Step 6.8.3.1

Apply the distributive property.

$- 2 \cdot 1 - 2 (- \cos^{2} (u)) - \cos^{2} (u) \sin^{2} (u) + 1 = 0$

Step 6.8.3.2

Multiply $- 2$ by $1$ .

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

Step 6.8.3.3

Multiply $- 1$ by $- 2$ .

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

Step 6.8.4

Add $- 2$ and $1$ .

$2 \cos^{2} (u) - \cos^{2} (u) \sin^{2} (u) - 1 = 0$

Step 6.8.5

Simplify the left side.

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Step 6.8.5.1

Simplify $2 \cos^{2} (u) - \cos^{2} (u) \sin^{2} (u) - 1$ .

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Step 6.8.5.1.1

Move $- 1$ .

$2 \cos^{2} (u) - 1 - \cos^{2} (u) \sin^{2} (u) = 0$

Step 6.8.5.1.2

Apply the cosine double-angle identity.

$\cos (2 u) - \cos^{2} (u) \sin^{2} (u) = 0$

Step 6.8.6

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

$1 - 2 \sin^{2} (u) - \cos^{2} (u) \sin^{2} (u) = 0$

Step 6.8.7

Subtract $1$ from both sides of the equation.

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

Step 6.8.8

Solve the equation for $u$ .

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Step 6.8.8.1

Add $1$ to both sides of the equation.

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

Step 6.8.8.2

Replace the $- 2 \sin^{2} (u)$ with $- 2 (1 - \cos^{2} (u))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

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

Step 6.8.8.3

Simplify each term.

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Step 6.8.8.3.1

Apply the distributive property.

$- 2 \cdot 1 - 2 (- \cos^{2} (u)) - \cos^{2} (u) \sin^{2} (u) + 1 = 0$

Step 6.8.8.3.2

Multiply $- 2$ by $1$ .

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

Step 6.8.8.3.3

Multiply $- 1$ by $- 2$ .

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

Step 6.8.8.4

Add $- 2$ and $1$ .

$2 \cos^{2} (u) - \cos^{2} (u) \sin^{2} (u) - 1 = 0$

Step 6.8.8.5

Simplify the left side.

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Step 6.8.8.5.1

Simplify $2 \cos^{2} (u) - \cos^{2} (u) \sin^{2} (u) - 1$ .

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Step 6.8.8.5.1.1

Move $- 1$ .

$2 \cos^{2} (u) - 1 - \cos^{2} (u) \sin^{2} (u) = 0$

Step 6.8.8.5.1.2

Apply the cosine double-angle identity.

$\cos (2 u) - \cos^{2} (u) \sin^{2} (u) = 0$

Step 6.8.8.6

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

$1 - 2 \sin^{2} (u) - \cos^{2} (u) \sin^{2} (u) = 0$

Step 6.8.8.7

Subtract $1$ from both sides of the equation.

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

Step 6.8.8.8

Solve the equation for $u$ .

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Step 6.8.8.8.1

Add $1$ to both sides of the equation.

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

Step 6.8.8.8.2

Replace the $- 2 \sin^{2} (u)$ with $- 2 (1 - \cos^{2} (u))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

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

Step 6.8.8.8.3

Simplify each term.

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Step 6.8.8.8.3.1

Apply the distributive property.

$- 2 \cdot 1 - 2 (- \cos^{2} (u)) - \cos^{2} (u) \sin^{2} (u) + 1 = 0$

Step 6.8.8.8.3.2

Multiply $- 2$ by $1$ .

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

Step 6.8.8.8.3.3

Multiply $- 1$ by $- 2$ .

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

Step 6.8.8.8.4

Add $- 2$ and $1$ .

$2 \cos^{2} (u) - \cos^{2} (u) \sin^{2} (u) - 1 = 0$

Step 6.8.8.8.5

Simplify the left side.

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Step 6.8.8.8.5.1

Simplify $2 \cos^{2} (u) - \cos^{2} (u) \sin^{2} (u) - 1$ .

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Step 6.8.8.8.5.1.1

Move $- 1$ .

$2 \cos^{2} (u) - 1 - \cos^{2} (u) \sin^{2} (u) = 0$

Step 6.8.8.8.5.1.2

Apply the cosine double-angle identity.

$\cos (2 u) - \cos^{2} (u) \sin^{2} (u) = 0$

Step 6.8.8.8.6

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

$\cos^{2} (u) - \sin^{2} (u) - \cos^{2} (u) \sin^{2} (u) = 0$

Step 6.8.8.8.7

Simplify the left side.

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Step 6.8.8.8.7.1

Simplify $\cos^{2} (u) - \sin^{2} (u) - \cos^{2} (u) \sin^{2} (u)$ .

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Step 6.8.8.8.7.1.1

Move $- \cos^{2} (u) \sin^{2} (u)$ .

$\cos^{2} (u) - \cos^{2} (u) \sin^{2} (u) - \sin^{2} (u) = 0$

Step 6.8.8.8.7.1.2

Multiply by $1$ .

$\cos^{2} (u) \cdot 1 - \cos^{2} (u) \sin^{2} (u) - \sin^{2} (u) = 0$

Step 6.8.8.8.7.1.3

Simplify with factoring out.

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Step 6.8.8.8.7.1.3.1

Factor $\cos^{2} (u)$ out of $- \cos^{2} (u) \sin^{2} (u)$ .

$\cos^{2} (u) \cdot 1 + \cos^{2} (u) (- 1 \sin^{2} (u)) - \sin^{2} (u) = 0$

Step 6.8.8.8.7.1.3.2

Factor $\cos^{2} (u)$ out of $\cos^{2} (u) \cdot 1 + \cos^{2} (u) (- 1 \sin^{2} (u))$ .

$\cos^{2} (u) \cdot (1 - 1 \sin^{2} (u)) - \sin^{2} (u) = 0$

Step 6.8.8.8.7.1.3.3

Rewrite $- 1 \sin^{2} (u)$ as $- \sin^{2} (u)$ .

$\cos^{2} (u) \cdot (1 - \sin^{2} (u)) - \sin^{2} (u) = 0$

Step 6.8.8.8.7.1.4

Apply pythagorean identity.

$\cos^{2} (u) \cdot \cos^{2} (u) - \sin^{2} (u) = 0$

Step 6.8.8.8.7.1.5

Rewrite $\cos^{2} (u) \cdot \cos^{2} (u)$ as ${(\cos (u) \cos (u))}^{2}$ .

${(\cos (u) \cos (u))}^{2} - \sin^{2} (u) = 0$

Step 6.8.8.8.7.1.6

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

$(\cos (u) \cos (u) + \sin (u)) (\cos (u) \cos (u) - \sin (u)) = 0$

Step 6.8.8.8.7.1.7

Multiply $\cos (u) \cos (u)$ .

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Step 6.8.8.8.7.1.7.1

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

$(\cos^{1} (u) \cos (u) + \sin (u)) (\cos (u) \cos (u) - \sin (u)) = 0$

Step 6.8.8.8.7.1.7.2

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

$(\cos^{1} (u) \cos^{1} (u) + \sin (u)) (\cos (u) \cos (u) - \sin (u)) = 0$

Step 6.8.8.8.7.1.7.3

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

$({\cos (u)}^{1 + 1} + \sin (u)) (\cos (u) \cos (u) - \sin (u)) = 0$

Step 6.8.8.8.7.1.7.4

Add $1$ and $1$ .

$(\cos^{2} (u) + \sin (u)) (\cos (u) \cos (u) - \sin (u)) = 0$

Step 6.8.8.8.7.1.8

Multiply $\cos (u) \cos (u)$ .

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Step 6.8.8.8.7.1.8.1

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

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

Step 6.8.8.8.7.1.8.2

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

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

Step 6.8.8.8.7.1.8.3

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

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

Step 6.8.8.8.7.1.8.4

Add $1$ and $1$ .

$(\cos^{2} (u) + \sin (u)) (\cos^{2} (u) - \sin (u)) = 0$

Step 6.8.8.8.7.1.9

Expand $(\cos^{2} (u) + \sin (u)) (\cos^{2} (u) - \sin (u))$ using the FOIL Method.

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Step 6.8.8.8.7.1.9.1

Apply the distributive property.

$\cos^{2} (u) (\cos^{2} (u) - \sin (u)) + \sin (u) (\cos^{2} (u) - \sin (u)) = 0$

Step 6.8.8.8.7.1.9.2

Apply the distributive property.

$\cos^{2} (u) \cos^{2} (u) + \cos^{2} (u) (- \sin (u)) + \sin (u) (\cos^{2} (u) - \sin (u)) = 0$

Step 6.8.8.8.7.1.9.3

Apply the distributive property.

$\cos^{2} (u) \cos^{2} (u) + \cos^{2} (u) (- \sin (u)) + \sin (u) \cos^{2} (u) + \sin (u) (- \sin (u)) = 0$

Step 6.8.8.8.7.1.10

Simplify terms.

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Step 6.8.8.8.7.1.10.1

Combine the opposite terms in $\cos^{2} (u) \cos^{2} (u) + \cos^{2} (u) (- \sin (u)) + \sin (u) \cos^{2} (u) + \sin (u) (- \sin (u))$ .

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Step 6.8.8.8.7.1.10.1.1

Reorder the factors in the terms $\cos^{2} (u) (- \sin (u))$ and $\sin (u) \cos^{2} (u)$ .

$\cos^{2} (u) \cos^{2} (u) - \cos^{2} (u) \sin (u) + \cos^{2} (u) \sin (u) + \sin (u) (- \sin (u)) = 0$

Step 6.8.8.8.7.1.10.1.2

Add $- \cos^{2} (u) \sin (u)$ and $\cos^{2} (u) \sin (u)$ .

$\cos^{2} (u) \cos^{2} (u) + 0 + \sin (u) (- \sin (u)) = 0$

Step 6.8.8.8.7.1.10.1.3

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

$\cos^{2} (u) \cos^{2} (u) + \sin (u) (- \sin (u)) = 0$

Step 6.8.8.8.7.1.10.2

Simplify each term.

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Step 6.8.8.8.7.1.10.2.1

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

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Step 6.8.8.8.7.1.10.2.1.1

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

${\cos (u)}^{2 + 2} + \sin (u) (- \sin (u)) = 0$

Step 6.8.8.8.7.1.10.2.1.2

Add $2$ and $2$ .

$\cos^{4} (u) + \sin (u) (- \sin (u)) = 0$

Step 6.8.8.8.7.1.10.2.2

Rewrite using the commutative property of multiplication.

$\cos^{4} (u) - \sin (u) \sin (u) = 0$

Step 6.8.8.8.7.1.10.2.3

Multiply $- \sin (u) \sin (u)$ .

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Step 6.8.8.8.7.1.10.2.3.1

Raise $\sin (u)$ to the power of $1$ .

$\cos^{4} (u) - (\sin^{1} (u) \sin (u)) = 0$

Step 6.8.8.8.7.1.10.2.3.2

Raise $\sin (u)$ to the power of $1$ .

$\cos^{4} (u) - (\sin^{1} (u) \sin^{1} (u)) = 0$

Step 6.8.8.8.7.1.10.2.3.3

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

$\cos^{4} (u) - {\sin (u)}^{1 + 1} = 0$

Step 6.8.8.8.7.1.10.2.3.4

Add $1$ and $1$ .

$\cos^{4} (u) - \sin^{2} (u) = 0$

Step 6.8.8.8.8

Solve the equation for $u$ .

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Step 6.8.8.8.8.1

Replace $\sin^{2} (u)$ with $1 - \cos^{2} (u)$ .

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

Step 6.8.8.8.8.2

Solve for $u$ .

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Step 6.8.8.8.8.2.1

Apply pythagorean identity.

$\cos^{4} (u) - \sin^{2} (u) = 0$

Step 6.8.8.8.8.2.2

Factor $\cos^{4} (u) - \sin^{2} (u)$ .

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Step 6.8.8.8.8.2.2.1

Rewrite $\cos^{4} (u)$ as ${(\cos^{2} (u))}^{2}$ .

${(\cos^{2} (u))}^{2} - \sin^{2} (u) = 0$

Step 6.8.8.8.8.2.2.2

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

$(\cos^{2} (u) + \sin (u)) (\cos^{2} (u) - \sin (u)) = 0$

Step 6.8.8.8.8.2.3

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

$\cos^{2} (u) + \sin (u) = 0$

$\cos^{2} (u) - \sin (u) = 0$

Step 6.8.8.8.8.2.4

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

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Step 6.8.8.8.8.2.4.1

Set $\cos^{2} (u) + \sin (u)$ equal to $0$ .

$\cos^{2} (u) + \sin (u) = 0$

Step 6.8.8.8.8.2.4.2

Solve $\cos^{2} (u) + \sin (u) = 0$ for $u$ .

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Step 6.8.8.8.8.2.4.2.1

Replace $\cos^{2} (u)$ with $1 - \sin^{2} (u)$ .

$(1 - \sin^{2} (u)) + \sin (u) = 0$

Step 6.8.8.8.8.2.4.2.2

Solve for $u$ .

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Step 6.8.8.8.8.2.4.2.2.1

Substitute $u$ for $\sin (u)$ .

$1 - {(u)}^{2} + u = 0$

Step 6.8.8.8.8.2.4.2.2.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6.8.8.8.8.2.4.2.2.3

Substitute the values $a = - 1$ , $b = 1$ , and $c = 1$ into the quadratic formula and solve for $u$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (- 1 \cdot 1)}}{2 \cdot - 1}$

Step 6.8.8.8.8.2.4.2.2.4

Simplify.

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Step 6.8.8.8.8.2.4.2.2.4.1

Simplify the numerator.

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Step 6.8.8.8.8.2.4.2.2.4.1.1

One to any power is one.

$u = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 1 \cdot 1}}{2 \cdot - 1}$

Step 6.8.8.8.8.2.4.2.2.4.1.2

Multiply $- 4 \cdot - 1 \cdot 1$ .

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Step 6.8.8.8.8.2.4.2.2.4.1.2.1

Multiply $- 4$ by $- 1$ .

$u = \frac{- 1 \pm \sqrt{1 + 4 \cdot 1}}{2 \cdot - 1}$

Step 6.8.8.8.8.2.4.2.2.4.1.2.2

Multiply $4$ by $1$ .

$u = \frac{- 1 \pm \sqrt{1 + 4}}{2 \cdot - 1}$

Step 6.8.8.8.8.2.4.2.2.4.1.3

Add $1$ and $4$ .

$u = \frac{- 1 \pm \sqrt{5}}{2 \cdot - 1}$

Step 6.8.8.8.8.2.4.2.2.4.2

Multiply $2$ by $- 1$ .

$u = \frac{- 1 \pm \sqrt{5}}{- 2}$

Step 6.8.8.8.8.2.4.2.2.4.3

Simplify $\frac{- 1 \pm \sqrt{5}}{- 2}$ .

$u = \frac{1 \pm \sqrt{5}}{2}$

Step 6.8.8.8.8.2.4.2.2.5

The final answer is the combination of both solutions.

$u = \frac{1 + \sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2}$

Step 6.8.8.8.8.2.4.2.2.6

Substitute $\sin (u)$ for $u$ .

$\sin (u) = \frac{1 + \sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2}$

Step 6.8.8.8.8.2.4.2.2.7

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

$\sin (u) = \frac{1 + \sqrt{5}}{2}$

$\sin (u) = \frac{1 - \sqrt{5}}{2}$

Step 6.8.8.8.8.2.4.2.2.8

Solve for $u$ in $\sin (u) = \frac{1 + \sqrt{5}}{2}$ .

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Step 6.8.8.8.8.2.4.2.2.8.1

The range of sine is $- 1 \leq y \leq 1$ . Since $\frac{1 + \sqrt{5}}{2}$ does not fall in this range, there is no solution.

No solution

Step 6.8.8.8.8.2.4.2.2.9

Solve for $u$ in $\sin (u) = \frac{1 - \sqrt{5}}{2}$ .

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Step 6.8.8.8.8.2.4.2.2.9.1

Take the inverse sine of both sides of the equation to extract $u$ from inside the sine.

$u = \arcsin (\frac{1 - \sqrt{5}}{2})$

Step 6.8.8.8.8.2.4.2.2.9.2

Simplify the right side.

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Step 6.8.8.8.8.2.4.2.2.9.2.1

Evaluate $\arcsin (\frac{1 - \sqrt{5}}{2})$ .

$u = - 0.66623943$

Step 6.8.8.8.8.2.4.2.2.9.3

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

$u = (3.14159265) + 0.66623943$

Step 6.8.8.8.8.2.4.2.2.9.4

Solve for $u$ .

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Step 6.8.8.8.8.2.4.2.2.9.4.1

Remove parentheses.

$u = 3.14159265 + 0.66623943$

Step 6.8.8.8.8.2.4.2.2.9.4.2

Remove parentheses.

$u = (3.14159265) + 0.66623943$

Step 6.8.8.8.8.2.4.2.2.9.4.3

Add $3.14159265$ and $0.66623943$ .

$u = 3.80783208$

Step 6.8.8.8.8.2.4.2.2.9.5

Find the period of $\sin (u)$ .

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Step 6.8.8.8.8.2.4.2.2.9.5.1

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

$\frac{2 π}{| b |}$

Step 6.8.8.8.8.2.4.2.2.9.5.2

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

$\frac{2 π}{| 1 |}$

Step 6.8.8.8.8.2.4.2.2.9.5.3

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

$\frac{2 π}{1}$

Step 6.8.8.8.8.2.4.2.2.9.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.8.8.8.8.2.4.2.2.9.6

Add $2 π$ to every negative angle to get positive angles.

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Step 6.8.8.8.8.2.4.2.2.9.6.1

Add $2 π$ to $- 0.66623943$ to find the positive angle.

$- 0.66623943 + 2 π$

Step 6.8.8.8.8.2.4.2.2.9.6.2

Subtract $0.66623943$ from $2 π$ .

$5.61694587$

Step 6.8.8.8.8.2.4.2.2.9.6.3

List the new angles.

$u = 5.61694587$

Step 6.8.8.8.8.2.4.2.2.9.7

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

$u = 3.80783208 + 2 π n, 5.61694587 + 2 π n$ , for any integer $n$

Step 6.8.8.8.8.2.4.2.2.10

List all of the solutions.

$u = 3.80783208 + 2 π n, 5.61694587 + 2 π n$ , for any integer $n$

Step 6.8.8.8.8.2.5

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

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Step 6.8.8.8.8.2.5.1

Set $\cos^{2} (u) - \sin (u)$ equal to $0$ .

$\cos^{2} (u) - \sin (u) = 0$

Step 6.8.8.8.8.2.5.2

Solve $\cos^{2} (u) - \sin (u) = 0$ for $u$ .

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Step 6.8.8.8.8.2.5.2.1

Replace $\cos^{2} (u)$ with $1 - \sin^{2} (u)$ .

$(1 - \sin^{2} (u)) - \sin (u) = 0$

Step 6.8.8.8.8.2.5.2.2

Solve for $u$ .

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Step 6.8.8.8.8.2.5.2.2.1

Substitute $u$ for $\sin (u)$ .

$1 - {(u)}^{2} - (u) = 0$

Step 6.8.8.8.8.2.5.2.2.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6.8.8.8.8.2.5.2.2.3

Substitute the values $a = - 1$ , $b = - 1$ , and $c = 1$ into the quadratic formula and solve for $u$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (- 1 \cdot 1)}}{2 \cdot - 1}$

Step 6.8.8.8.8.2.5.2.2.4

Simplify.

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Step 6.8.8.8.8.2.5.2.2.4.1

Simplify the numerator.

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Step 6.8.8.8.8.2.5.2.2.4.1.1

Raise $- 1$ to the power of $2$ .

$u = \frac{1 \pm \sqrt{1 - 4 \cdot - 1 \cdot 1}}{2 \cdot - 1}$

Step 6.8.8.8.8.2.5.2.2.4.1.2

Multiply $- 4 \cdot - 1 \cdot 1$ .

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Step 6.8.8.8.8.2.5.2.2.4.1.2.1

Multiply $- 4$ by $- 1$ .

$u = \frac{1 \pm \sqrt{1 + 4 \cdot 1}}{2 \cdot - 1}$

Step 6.8.8.8.8.2.5.2.2.4.1.2.2

Multiply $4$ by $1$ .

$u = \frac{1 \pm \sqrt{1 + 4}}{2 \cdot - 1}$

Step 6.8.8.8.8.2.5.2.2.4.1.3

Add $1$ and $4$ .

$u = \frac{1 \pm \sqrt{5}}{2 \cdot - 1}$

Step 6.8.8.8.8.2.5.2.2.4.2

Multiply $2$ by $- 1$ .

$u = \frac{1 \pm \sqrt{5}}{- 2}$

Step 6.8.8.8.8.2.5.2.2.4.3

Move the negative in front of the fraction.

$u = - \frac{1 \pm \sqrt{5}}{2}$

Step 6.8.8.8.8.2.5.2.2.5

The final answer is the combination of both solutions.

$u = - \frac{1 + \sqrt{5}}{2}, - \frac{1 - \sqrt{5}}{2}$

Step 6.8.8.8.8.2.5.2.2.6

Substitute $\sin (u)$ for $u$ .

$\sin (u) = - \frac{1 + \sqrt{5}}{2}, - \frac{1 - \sqrt{5}}{2}$

Step 6.8.8.8.8.2.5.2.2.7

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

$\sin (u) = - \frac{1 + \sqrt{5}}{2}$

$\sin (u) = - \frac{1 - \sqrt{5}}{2}$

Step 6.8.8.8.8.2.5.2.2.8

Solve for $u$ in $\sin (u) = - \frac{1 + \sqrt{5}}{2}$ .

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Step 6.8.8.8.8.2.5.2.2.8.1

The range of sine is $- 1 \leq y \leq 1$ . Since $- \frac{1 + \sqrt{5}}{2}$ does not fall in this range, there is no solution.

No solution

Step 6.8.8.8.8.2.5.2.2.9

Solve for $u$ in $\sin (u) = - \frac{1 - \sqrt{5}}{2}$ .

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Step 6.8.8.8.8.2.5.2.2.9.1

Take the inverse sine of both sides of the equation to extract $u$ from inside the sine.

$u = \arcsin (- \frac{1 - \sqrt{5}}{2})$

Step 6.8.8.8.8.2.5.2.2.9.2

Simplify the right side.

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Step 6.8.8.8.8.2.5.2.2.9.2.1

Evaluate $\arcsin (- \frac{1 - \sqrt{5}}{2})$ .

$u = 0.66623943$

Step 6.8.8.8.8.2.5.2.2.9.3

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$u = 2 (3.14159265) - 0.66623943 + 3.14159265$

Step 6.8.8.8.8.2.5.2.2.9.4

Simplify the expression to find the second solution.

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Step 6.8.8.8.8.2.5.2.2.9.4.1

Subtract $2 π$ from $2 (3.14159265) - 0.66623943 + 3.14159265$ .

$u = 2 (3.14159265) - 0.66623943 + 3.14159265 - 2 π$

Step 6.8.8.8.8.2.5.2.2.9.4.2

The resulting angle of $2.47535322$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) - 0.66623943 + 3.14159265$ .

$u = 2.47535322$

Step 6.8.8.8.8.2.5.2.2.9.5

Find the period of $\sin (u)$ .

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Step 6.8.8.8.8.2.5.2.2.9.5.1

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

$\frac{2 π}{| b |}$

Step 6.8.8.8.8.2.5.2.2.9.5.2

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

$\frac{2 π}{| 1 |}$

Step 6.8.8.8.8.2.5.2.2.9.5.3

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

$\frac{2 π}{1}$

Step 6.8.8.8.8.2.5.2.2.9.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.8.8.8.8.2.5.2.2.9.6

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

$u = 0.66623943 + 2 π n, 2.47535322 + 2 π n$ , for any integer $n$

Step 6.8.8.8.8.2.5.2.2.10

List all of the solutions.

$u = 0.66623943 + 2 π n, 2.47535322 + 2 π n$ , for any integer $n$

Step 6.8.8.8.8.2.6

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

$u = 3.80783208 + 2 π n, 5.61694587 + 2 π n, 0.66623943 + 2 π n, 2.47535322 + 2 π n$ , for any integer $n$

Step 7

Consolidate the answers.

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Step 7.1

Consolidate $3.80783208 + 2 π n$ and $0.66623943 + 2 π n$ to $0.66623943 + π n$ .

$u = 0.66623943 + π n, 5.61694587 + 2 π n, 2.47535322 + 2 π n$ , for any integer $n$

Step 7.2

Consolidate $5.61694587 + 2 π n$ and $2.47535322 + 2 π n$ to $2.47535322 + π n$ .

$u = 0.66623943 + π n, 2.47535322 + π n$ , for any integer $n$