Precalculus Examples

$\sec (x) \csc (x) = 2 \sec (x)$

Step 1

Subtract $2 \sec (x)$ from both sides of the equation.

$\sec (x) \csc (x) - 2 \sec (x) = 0$

Step 2

Factor $\sec (x)$ out of $\sec (x) \csc (x) - 2 \sec (x)$ .

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Step 2.1

Factor $\sec (x)$ out of $- 2 \sec (x)$ .

$\sec (x) \csc (x) + \sec (x) \cdot - 2 = 0$

Step 2.2

Factor $\sec (x)$ out of $\sec (x) \csc (x) + \sec (x) \cdot - 2$ .

$\sec (x) (\csc (x) - 2) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\sec (x) = 0$

$\csc (x) - 2 = 0$

Step 4

Set $\sec (x)$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $\sec (x)$ equal to $0$ .

$\sec (x) = 0$

Step 4.2

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 $\csc (x) - 2$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $\csc (x) - 2$ equal to $0$ .

$\csc (x) - 2 = 0$

Step 5.2

Solve $\csc (x) - 2 = 0$ for $x$ .

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Step 5.2.1

Add $2$ to both sides of the equation.

$\csc (x) = 2$

Step 5.2.2

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

$x = arccsc (2)$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

The exact value of $arccsc (2)$ is $\frac{π}{6}$ .

$x = \frac{π}{6}$

Step 5.2.4

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.

$x = π - \frac{π}{6}$

Step 5.2.5

Simplify $π - \frac{π}{6}$ .

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Step 5.2.5.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 5.2.5.2

Combine fractions.

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Step 5.2.5.2.1

Combine $π$ and $\frac{6}{6}$ .

$x = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 5.2.5.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 6 - π}{6}$

Step 5.2.5.3

Simplify the numerator.

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Step 5.2.5.3.1

Move $6$ to the left of $π$ .

$x = \frac{6 \cdot π - π}{6}$

Step 5.2.5.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{6}$

Step 5.2.6

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

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Step 5.2.6.1

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

$\frac{2 π}{| b |}$

Step 5.2.6.2

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

$\frac{2 π}{| 1 |}$

Step 5.2.6.3

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

$\frac{2 π}{1}$

Step 5.2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.7

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

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$

Step 6

The final solution is all the values that make $\sec (x) (\csc (x) - 2) = 0$ true.

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$