Precalculus Examples

$\frac{\sec (x)}{\csc (x)}$

Step 1

Set the denominator in $\frac{\sec (x)}{\csc (x)}$ equal to $0$ to find where the expression is undefined.

$\csc (x) = 0$

Step 2

The range of cosecant is $y \leq - 1$ and $y \geq 1$ . Since $0$ does not fall in this range, there is no solution.

No solution

Step 3

Set the argument in $\sec (x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

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

Step 4

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

$x = π n$ , for any integer $n$

Step 5

The domain is all values of $x$ that make the expression defined.

Set-Builder Notation:

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

Step 6