Trigonometry Examples

$\frac{\sec^{2} (t) - 1}{\tan^{2} (t)}$

Step 1

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

$\tan^{2} (t) = 0$

Step 2

Solve for $t$ .

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Step 2.1

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

$\tan (t) = \pm \sqrt{0}$

Step 2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.2.1

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

$\tan (t) = \pm \sqrt{0^{2}}$

Step 2.2.2

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

$\tan (t) = \pm 0$

Step 2.2.3

Plus or minus $0$ is $0$ .

$\tan (t) = 0$

Step 2.3

Take the inverse tangent of both sides of the equation to extract $t$ from inside the tangent.

$t = \arctan (0)$

Step 2.4

Simplify the right side.

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Step 2.4.1

The exact value of $\arctan (0)$ is $0$ .

$t = 0$

Step 2.5

The tangent 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.

$t = π + 0$

Step 2.6

Add $π$ and $0$ .

$t = π$

Step 2.7

Find the period of $\tan (t)$ .

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Step 2.7.1

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

$\frac{π}{| b |}$

Step 2.7.2

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

$\frac{π}{| 1 |}$

Step 2.7.3

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

$\frac{π}{1}$

Step 2.7.4

Divide $π$ by $1$ .

$π$

Step 2.8

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

$t = π n, π + π n$ , for any integer $n$

Step 2.9

Consolidate the answers.

$t = π n$ , for any integer $n$

Step 3

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

$t = \frac{π}{2} + π n$ , 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 $0$ .

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

Step 5