Trigonometry Examples

$\frac{\cot^{2} (x)}{\csc (x) - 1}$

Step 1

Set the denominator in $\frac{\cot^{2} (x)}{\csc (x) - 1}$ equal to $0$ to find where the expression is undefined.

$\csc (x) - 1 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $1$ to both sides of the equation.

$\csc (x) = 1$

Step 2.2

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

$x = arccsc (1)$

Step 2.3

Simplify the right side.

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Step 2.3.1

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

$x = \frac{π}{2}$

Step 2.4

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 2.5

Simplify $π - \frac{π}{2}$ .

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Step 2.5.1

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 2.5.2

Combine fractions.

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Step 2.5.2.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 2.5.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 - π}{2}$

Step 2.5.3

Simplify the numerator.

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Step 2.5.3.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π - π}{2}$

Step 2.5.3.2

Subtract $π$ from $2 π$ .

$x = \frac{π}{2}$

Step 2.6

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

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Step 2.6.1

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

$\frac{2 π}{| b |}$

Step 2.6.2

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

$\frac{2 π}{| 1 |}$

Step 2.6.3

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

$\frac{2 π}{1}$

Step 2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.7

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 $n$

Step 3

Set the argument in $\cot (x)$ equal to $π n$ to find where the expression is undefined.

$x = π n$ , for any integer $n$

Step 4

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

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

Step 5