Precalculus Examples

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

Step 1

Subtract $2 \cot (x)$ from both sides of the equation.

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

Step 2

Simplify the left side of the equation.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite $\cot (x)$ in terms of sines and cosines.

$\frac{\cos (x)}{\sin (x)} \cdot \cos^{2} (x) - 2 \cot (x) = 0$

Step 2.1.2

Multiply $\frac{\cos (x)}{\sin (x)} \cos^{2} (x)$ .

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Step 2.1.2.1

Combine $\frac{\cos (x)}{\sin (x)}$ and $\cos^{2} (x)$ .

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

Step 2.1.2.2

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

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Step 2.1.2.2.1

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

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Step 2.1.2.2.1.1

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

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

Step 2.1.2.2.1.2

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

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

Step 2.1.2.2.2

Add $1$ and $2$ .

$\frac{\cos^{3} (x)}{\sin (x)} - 2 \cot (x) = 0$

Step 2.1.3

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 2.1.4

Combine $- 2$ and $\frac{\cos (x)}{\sin (x)}$ .

$\frac{\cos^{3} (x)}{\sin (x)} + \frac{- 2 \cos (x)}{\sin (x)} = 0$

Step 2.1.5

Move the negative in front of the fraction.

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

Step 2.2

Simplify each term.

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Step 2.2.1

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

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

Step 2.2.2

Separate fractions.

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

Step 2.2.3

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\frac{\cos^{2} (x)}{1} \cdot \cot (x) - \frac{2 \cos (x)}{\sin (x)} = 0$

Step 2.2.4

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

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

Step 2.2.5

Separate fractions.

$\cos^{2} (x) \cot (x) - (\frac{2}{1} \cdot \frac{\cos (x)}{\sin (x)}) = 0$

Step 2.2.6

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\cos^{2} (x) \cot (x) - (\frac{2}{1} \cdot \cot (x)) = 0$

Step 2.2.7

Divide $2$ by $1$ .

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

Step 2.2.8

Multiply $2$ by $- 1$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Factor $\cot (x)$ out of $- 2 \cot (x)$ .

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

Step 3.3

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

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

Step 4

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

$\cot (x) = 0$

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

Step 5

Set $\cot (x)$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $\cot (x)$ equal to $0$ .

$\cot (x) = 0$

Step 5.2

Solve $\cot (x) = 0$ for $x$ .

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Step 5.2.1

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

$x = arccot (0)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

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

$x = \frac{π}{2}$

Step 5.2.3

The cotangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = π + \frac{π}{2}$

Step 5.2.4

Simplify $π + \frac{π}{2}$ .

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Step 5.2.4.1

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

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

Step 5.2.4.2

Combine fractions.

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Step 5.2.4.2.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 5.2.4.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 + π}{2}$

Step 5.2.4.3

Simplify the numerator.

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Step 5.2.4.3.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π + π}{2}$

Step 5.2.4.3.2

Add $2 π$ and $π$ .

$x = \frac{3 π}{2}$

Step 5.2.5

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

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Step 5.2.5.1

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

$\frac{π}{| b |}$

Step 5.2.5.2

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

$\frac{π}{| 1 |}$

Step 5.2.5.3

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

$\frac{π}{1}$

Step 5.2.5.4

Divide $π$ by $1$ .

$π$

Step 5.2.6

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

Add $2$ to both sides of the equation.

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

Step 6.2.2

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

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

Step 6.2.3

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

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Step 6.2.3.1

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

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

Step 6.2.3.2

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

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

Step 6.2.3.3

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

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

Step 6.2.4

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

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

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

Step 6.2.5

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

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Step 6.2.5.1

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

No solution

Step 6.2.6

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

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Step 6.2.6.1

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

No solution

Step 7

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

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

Step 8

Consolidate the answers.

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