Algebra Examples

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

Step 1

Rewrite $\sin (θ) \cot (θ) - \cos^{2} (θ)$ as a difference of squares.

${\sqrt{\cos (θ)}}^{2} - \cos^{2} (θ)$

Step 2

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

$(\sqrt{\cos (θ)} + \cos (θ)) (\sqrt{\cos (θ)} - \cos (θ))$

Step 3

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

$\sqrt{\cos (θ)} + \cos (θ) = 0$

$\sqrt{\cos (θ)} - \cos (θ) = 0$

Step 4

Set $\sqrt{\cos (θ)} + \cos (θ)$ equal to $0$ and solve for $θ$ .

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Step 4.1

Set $\sqrt{\cos (θ)} + \cos (θ)$ equal to $0$ .

$\sqrt{\cos (θ)} + \cos (θ) = 0$

Step 4.2

Solve $\sqrt{\cos (θ)} + \cos (θ) = 0$ for $θ$ .

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Step 4.2.1

Subtract $\cos (θ)$ from both sides of the equation.

$\sqrt{\cos (θ)} = - \cos (θ)$

Step 4.2.2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{\cos (θ)}}^{2} = {(- \cos (θ))}^{2}$

Step 4.2.3

Simplify each side of the equation.

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Step 4.2.3.1

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

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

Step 4.2.3.2

Simplify the left side.

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Step 4.2.3.2.1

Simplify ${({\cos (θ)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.3.2.1.1

Multiply the exponents in ${({\cos (θ)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.3.2.1.1.1

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

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

Step 4.2.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.3.2.1.1.2.1

Cancel the common factor.

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

Step 4.2.3.2.1.1.2.2

Rewrite the expression.

$\cos^{1} (θ) = {(- \cos (θ))}^{2}$

Step 4.2.3.2.1.2

Simplify.

$\cos (θ) = {(- \cos (θ))}^{2}$

Step 4.2.3.3

Simplify the right side.

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Step 4.2.3.3.1

Simplify ${(- \cos (θ))}^{2}$ .

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Step 4.2.3.3.1.1

Apply the product rule to $- \cos (θ)$ .

$\cos (θ) = {(- 1)}^{2} \cos^{2} (θ)$

Step 4.2.3.3.1.2

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

$\cos (θ) = 1 \cos^{2} (θ)$

Step 4.2.3.3.1.3

Multiply $\cos^{2} (θ)$ by $1$ .

$\cos (θ) = \cos^{2} (θ)$

Step 4.2.4

Solve for $\cos (θ)$ .

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Step 4.2.4.1

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

$\cos (θ) - \cos^{2} (θ) = 0$

Step 4.2.4.2

Factor the left side of the equation.

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Step 4.2.4.2.1

Let $u = \cos (θ)$ . Substitute $u$ for all occurrences of $\cos (θ)$ .

$u - u^{2} = 0$

Step 4.2.4.2.2

Factor $u$ out of $u - u^{2}$ .

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Step 4.2.4.2.2.1

Raise $u$ to the power of $1$ .

$u - u^{2} = 0$

Step 4.2.4.2.2.2

Factor $u$ out of $u^{1}$ .

$u \cdot 1 - u^{2} = 0$

Step 4.2.4.2.2.3

Factor $u$ out of $- u^{2}$ .

$u \cdot 1 + u (- u) = 0$

Step 4.2.4.2.2.4

Factor $u$ out of $u \cdot 1 + u (- u)$ .

$u (1 - u) = 0$

Step 4.2.4.2.3

Replace all occurrences of $u$ with $\cos (θ)$ .

$\cos (θ) (1 - \cos (θ)) = 0$

Step 4.2.4.3

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

$\cos (θ) = 0$

$1 - \cos (θ) = 0$

Step 4.2.4.4

Set $\cos (θ)$ equal to $0$ .

$\cos (θ) = 0$

Step 4.2.4.5

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

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Step 4.2.4.5.1

Set $1 - \cos (θ)$ equal to $0$ .

$1 - \cos (θ) = 0$

Step 4.2.4.5.2

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

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Step 4.2.4.5.2.1

Subtract $1$ from both sides of the equation.

$- \cos (θ) = - 1$

Step 4.2.4.5.2.2

Divide each term in $- \cos (θ) = - 1$ by $- 1$ and simplify.

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Step 4.2.4.5.2.2.1

Divide each term in $- \cos (θ) = - 1$ by $- 1$ .

$\frac{- \cos (θ)}{- 1} = \frac{- 1}{- 1}$

Step 4.2.4.5.2.2.2

Simplify the left side.

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Step 4.2.4.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{\cos (θ)}{1} = \frac{- 1}{- 1}$

Step 4.2.4.5.2.2.2.2

Divide $\cos (θ)$ by $1$ .

$\cos (θ) = \frac{- 1}{- 1}$

Step 4.2.4.5.2.2.3

Simplify the right side.

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Step 4.2.4.5.2.2.3.1

Divide $- 1$ by $- 1$ .

$\cos (θ) = 1$

Step 4.2.4.6

The final solution is all the values that make $\cos (θ) (1 - \cos (θ)) = 0$ true.

$\cos (θ) = 0, 1$

Step 4.2.5

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

$\cos (θ) = 0$

$\cos (θ) = 1$

Step 4.2.6

Solve for $θ$ in $\cos (θ) = 0$ .

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Step 4.2.6.1

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

$θ = \arccos (0)$

Step 4.2.6.2

Simplify the right side.

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Step 4.2.6.2.1

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

$θ = \frac{π}{2}$

Step 4.2.6.3

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.

$θ = 2 π - \frac{π}{2}$

Step 4.2.6.4

Simplify $2 π - \frac{π}{2}$ .

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Step 4.2.6.4.1

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

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

Step 4.2.6.4.2

Combine fractions.

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Step 4.2.6.4.2.1

Combine $2 π$ and $\frac{2}{2}$ .

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

Step 4.2.6.4.2.2

Combine the numerators over the common denominator.

$θ = \frac{2 π \cdot 2 - π}{2}$

Step 4.2.6.4.3

Simplify the numerator.

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Step 4.2.6.4.3.1

Multiply $2$ by $2$ .

$θ = \frac{4 π - π}{2}$

Step 4.2.6.4.3.2

Subtract $π$ from $4 π$ .

$θ = \frac{3 π}{2}$

Step 4.2.6.5

Find the period of $\cos (θ)$ .

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Step 4.2.6.5.1

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

$\frac{2 π}{| b |}$

Step 4.2.6.5.2

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

$\frac{2 π}{| 1 |}$

Step 4.2.6.5.3

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

$\frac{2 π}{1}$

Step 4.2.6.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.2.6.6

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

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

Step 4.2.7

Solve for $θ$ in $\cos (θ) = 1$ .

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Step 4.2.7.1

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

$θ = \arccos (1)$

Step 4.2.7.2

Simplify the right side.

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Step 4.2.7.2.1

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

$θ = 0$

Step 4.2.7.3

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.

$θ = 2 π - 0$

Step 4.2.7.4

Subtract $0$ from $2 π$ .

$θ = 2 π$

Step 4.2.7.5

Find the period of $\cos (θ)$ .

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Step 4.2.7.5.1

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

$\frac{2 π}{| b |}$

Step 4.2.7.5.2

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

$\frac{2 π}{| 1 |}$

Step 4.2.7.5.3

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

$\frac{2 π}{1}$

Step 4.2.7.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.2.7.6

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

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

Step 4.2.8

List all of the solutions.

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

Step 4.2.9

Consolidate the solutions.

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Step 4.2.9.1

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

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

Step 4.2.9.2

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

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

Step 5

The final solution is all the values that make $(\sqrt{\cos (θ)} + \cos (θ)) (\sqrt{\cos (θ)} - \cos (θ)) = 0$ true.

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