Trigonometry Examples

$\frac{1}{\csc^{2} (x)} + \frac{1}{\sec^{2} (x)}$

Step 1

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

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

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

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

$\csc (x) = \pm \sqrt{0}$

Step 2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 2.2.1

Rewrite $0$ as $0^{2}$ .

$\csc (x) = \pm \sqrt{0^{2}}$

Step 2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$\csc (x) = \pm 0$

Step 2.2.3

Plus or minus $0$ is $0$ .

$\csc (x) = 0$

Step 2.3

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

$\sec^{2} (x) = 0$

Step 4

Solve for $x$ .

Tap for more steps...

Step 4.1

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

$\sec (x) = \pm \sqrt{0}$

Step 4.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 4.2.1

Rewrite $0$ as $0^{2}$ .

$\sec (x) = \pm \sqrt{0^{2}}$

Step 4.2.2

Pull terms out from under the radical, assuming positive real numbers.

$\sec (x) = \pm 0$

Step 4.2.3

Plus or minus $0$ is $0$ .

$\sec (x) = 0$

Step 4.3

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

No solution

Step 5

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 6

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 7

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 $0$ .

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

Step 8