Precalculus Examples

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

Step 1

Replace the $\csc^{2} (x)$ with $\cot^{2} (x) + 1$ based on the $\cot^{2} (x) + 1 = \csc^{2} (x)$ identity.

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

Step 2

Reorder the polynomial.

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

Step 3

Substitute $u$ for $\cot (x)$ .

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

Step 4

Factor using the perfect square rule.

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Step 4.1

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

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

Step 4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 u = 2 \cdot u \cdot 1$

Step 4.3

Rewrite the polynomial.

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

Step 4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 1$ .

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

Step 5

Set the $u - 1$ equal to $0$ .

$u - 1 = 0$

Step 6

Add $1$ to both sides of the equation.

$u = 1$

Step 7

Substitute $\cot (x)$ for $u$ .

$\cot (x) = 1$

Step 8

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

$x = arccot (1)$

Step 9

Simplify the right side.

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Step 9.1

The exact value of $arccot (1)$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 10

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.

$x = π + \frac{π}{4}$

Step 11

Simplify $π + \frac{π}{4}$ .

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Step 11.1

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

$x = π \cdot \frac{4}{4} + \frac{π}{4}$

Step 11.2

Combine fractions.

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Step 11.2.1

Combine $π$ and $\frac{4}{4}$ .

$x = \frac{π \cdot 4}{4} + \frac{π}{4}$

Step 11.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Step 11.3

Simplify the numerator.

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Step 11.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{4}$

Step 11.3.2

Add $4 π$ and $π$ .

$x = \frac{5 π}{4}$

Step 12

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

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Step 12.1

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

$\frac{π}{| b |}$

Step 12.2

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

$\frac{π}{| 1 |}$

Step 12.3

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

$\frac{π}{1}$

Step 12.4

Divide $π$ by $1$ .

$π$

Step 13

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

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

Step 14

Consolidate the answers.

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