Trigonometry Examples

$\frac{1}{\cos^{2} (l)} - \tan^{2} (l)$

Step 1

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

$\cos^{2} (l) = 0$

Step 2

Solve for $l$ .

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

$\cos (l) = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

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

$\cos (l) = \pm \sqrt{0^{2}}$

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

$\cos (l) = \pm 0$

Plus or minus $0$ is $0$ .

$\cos (l) = 0$

Take the inverse cosine of both sides of the equation to extract $l$ from inside the cosine.

$l = \arccos (0)$

Simplify the right side.

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The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$l = \frac{π}{2}$

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$l = 2 π - \frac{π}{2}$

Simplify $2 π - \frac{π}{2}$ .

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To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Combine fractions.

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Combine $2 π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

$l = \frac{2 π \cdot 2 - π}{2}$

Simplify the numerator.

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Multiply $2$ by $2$ .

$l = \frac{4 π - π}{2}$

Subtract $π$ from $4 π$ .

$l = \frac{3 π}{2}$

Find the period of $\cos (l)$ .

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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

$l = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Consolidate the answers.

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

Step 3

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 $0$ .

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

Step 4