Precalculus Examples

$\cos (x) = \cot (x)$

Step 1

Divide each term in $\cos (x) = \cot (x)$ by $\cos (x)$ and simplify.

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Step 1.1

Divide each term in $\cos (x) = \cot (x)$ by $\cos (x)$ .

$\frac{\cos (x)}{\cos (x)} = \frac{\cot (x)}{\cos (x)}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $\cos (x)$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{\cos (x)}{\cos (x)} = \frac{\cot (x)}{\cos (x)}$

Step 1.2.1.2

Rewrite the expression.

$1 = \frac{\cot (x)}{\cos (x)}$

Step 1.3

Simplify the right side.

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Step 1.3.1

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

$1 = \frac{\frac{\cos (x)}{\sin (x)}}{\cos (x)}$

Step 1.3.2

Rewrite $\frac{\frac{\cos (x)}{\sin (x)}}{\cos (x)}$ as a product.

$1 = \frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\cos (x)}$

Step 1.3.3

Cancel the common factor of $\cos (x)$ .

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Step 1.3.3.1

Cancel the common factor.

$1 = \frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\cos (x)}$

Step 1.3.3.2

Rewrite the expression.

$1 = \frac{1}{\sin (x)}$

Step 1.3.4

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$1 = \csc (x)$

Step 2

Rewrite the equation as $\csc (x) = 1$ .

$\csc (x) = 1$

Step 3

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

$x = arccsc (1)$

Step 4

Simplify the right side.

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Step 4.1

The exact value of $arccsc (1)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 5

The cosecant 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.

$x = π - \frac{π}{2}$

Step 6

Simplify $π - \frac{π}{2}$ .

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Step 6.1

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 6.2

Combine fractions.

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Step 6.2.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 6.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 - π}{2}$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π - π}{2}$

Step 6.3.2

Subtract $π$ from $2 π$ .

$x = \frac{π}{2}$

Step 7

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

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

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

$\frac{2 π}{| 1 |}$

Step 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 7.4

Divide $2 π$ by $1$ .

$2 π$

Step 8

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

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