Algebra Examples

$\tan^{2} (x) - \sec (x) = 1$

Step 1

Rewrite $\tan^{2} (x) - \sec (x)$ as a difference of squares.

$\tan^{2} (x) - {\sqrt{\sec (x)}}^{2}$

Step 2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \tan (x)$ and $b = \sqrt{\sec (x)}$ .

$(\tan (x) + \sqrt{\sec (x)}) (\tan (x) - \sqrt{\sec (x)})$

Step 3

Solve for $\sqrt{\sec (x)}$ .

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $(\tan (x) + \sqrt{\sec (x)}) (\tan (x) - \sqrt{\sec (x)})$ .

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Step 3.1.1.1

Expand $(\tan (x) + \sqrt{\sec (x)}) (\tan (x) - \sqrt{\sec (x)})$ using the FOIL Method.

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Step 3.1.1.1.1

Apply the distributive property.

$\tan (x) (\tan (x) - \sqrt{\sec (x)}) + \sqrt{\sec (x)} (\tan (x) - \sqrt{\sec (x)}) = 1$

Step 3.1.1.1.2

Apply the distributive property.

$\tan (x) \tan (x) + \tan (x) (- \sqrt{\sec (x)}) + \sqrt{\sec (x)} (\tan (x) - \sqrt{\sec (x)}) = 1$

Step 3.1.1.1.3

Apply the distributive property.

$\tan (x) \tan (x) + \tan (x) (- \sqrt{\sec (x)}) + \sqrt{\sec (x)} \tan (x) + \sqrt{\sec (x)} (- \sqrt{\sec (x)}) = 1$

Step 3.1.1.2

Simplify terms.

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Step 3.1.1.2.1

Combine the opposite terms in $\tan (x) \tan (x) + \tan (x) (- \sqrt{\sec (x)}) + \sqrt{\sec (x)} \tan (x) + \sqrt{\sec (x)} (- \sqrt{\sec (x)})$ .

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Step 3.1.1.2.1.1

Reorder the factors in the terms $\tan (x) (- \sqrt{\sec (x)})$ and $\sqrt{\sec (x)} \tan (x)$ .

$\tan (x) \tan (x) - \sqrt{\sec (x)} \tan (x) + \sqrt{\sec (x)} \tan (x) + \sqrt{\sec (x)} (- \sqrt{\sec (x)}) = 1$

Step 3.1.1.2.1.2

Add $- \sqrt{\sec (x)} \tan (x)$ and $\sqrt{\sec (x)} \tan (x)$ .

$\tan (x) \tan (x) + 0 + \sqrt{\sec (x)} (- \sqrt{\sec (x)}) = 1$

Step 3.1.1.2.1.3

Add $\tan (x) \tan (x)$ and $0$ .

$\tan (x) \tan (x) + \sqrt{\sec (x)} (- \sqrt{\sec (x)}) = 1$

Step 3.1.1.2.2

Simplify each term.

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Step 3.1.1.2.2.1

Multiply $\tan (x) \tan (x)$ .

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Step 3.1.1.2.2.1.1

Raise $\tan (x)$ to the power of $1$ .

$\tan^{1} (x) \tan (x) + \sqrt{\sec (x)} (- \sqrt{\sec (x)}) = 1$

Step 3.1.1.2.2.1.2

Raise $\tan (x)$ to the power of $1$ .

$\tan^{1} (x) \tan^{1} (x) + \sqrt{\sec (x)} (- \sqrt{\sec (x)}) = 1$

Step 3.1.1.2.2.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${\tan (x)}^{1 + 1} + \sqrt{\sec (x)} (- \sqrt{\sec (x)}) = 1$

Step 3.1.1.2.2.1.4

Add $1$ and $1$ .

$\tan^{2} (x) + \sqrt{\sec (x)} (- \sqrt{\sec (x)}) = 1$

Step 3.1.1.2.2.2

Rewrite using the commutative property of multiplication.

$\tan^{2} (x) - \sqrt{\sec (x)} \sqrt{\sec (x)} = 1$

Step 3.1.1.2.2.3

Multiply $\sqrt{\sec (x)}$ by $\sqrt{\sec (x)}$ by adding the exponents.

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Step 3.1.1.2.2.3.1

Move $\sqrt{\sec (x)}$ .

$\tan^{2} (x) - (\sqrt{\sec (x)} \sqrt{\sec (x)}) = 1$

Step 3.1.1.2.2.3.2

Multiply $\sqrt{\sec (x)}$ by $\sqrt{\sec (x)}$ .

$\tan^{2} (x) - {\sqrt{\sec (x)}}^{2} = 1$

Step 3.2

Subtract $\tan^{2} (x)$ from both sides of the equation.

$- {\sqrt{\sec (x)}}^{2} = 1 - \tan^{2} (x)$

Step 3.3

Divide each term in $- {\sqrt{\sec (x)}}^{2} = 1 - \tan^{2} (x)$ by $- 1$ and simplify.

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Step 3.3.1

Divide each term in $- {\sqrt{\sec (x)}}^{2} = 1 - \tan^{2} (x)$ by $- 1$ .

$\frac{- {\sqrt{\sec (x)}}^{2}}{- 1} = \frac{1}{- 1} + \frac{- \tan^{2} (x)}{- 1}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Dividing two negative values results in a positive value.

$\frac{{\sqrt{\sec (x)}}^{2}}{1} = \frac{1}{- 1} + \frac{- \tan^{2} (x)}{- 1}$

Step 3.3.2.2

Divide ${\sqrt{\sec (x)}}^{2}$ by $1$ .

${\sqrt{\sec (x)}}^{2} = \frac{1}{- 1} + \frac{- \tan^{2} (x)}{- 1}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Divide $1$ by $- 1$ .

${\sqrt{\sec (x)}}^{2} = - 1 + \frac{- \tan^{2} (x)}{- 1}$

Step 3.3.3.1.2

Dividing two negative values results in a positive value.

${\sqrt{\sec (x)}}^{2} = - 1 + \frac{\tan^{2} (x)}{1}$

Step 3.3.3.1.3

Divide $\tan^{2} (x)$ by $1$ .

${\sqrt{\sec (x)}}^{2} = - 1 + \tan^{2} (x)$

Step 3.4

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

$\sqrt{\sec (x)} = \pm \sqrt{- 1 + \tan^{2} (x)}$

Step 3.5

Simplify $\pm \sqrt{- 1 + \tan^{2} (x)}$ .

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Step 3.5.1

Simplify the expression.

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Step 3.5.1.1

Rewrite $1$ as $1^{2}$ .

$\sqrt{\sec (x)} = \pm \sqrt{- 1^{2} + \tan^{2} (x)}$

Step 3.5.1.2

Reorder $- 1^{2}$ and $\tan^{2} (x)$ .

$\sqrt{\sec (x)} = \pm \sqrt{\tan^{2} (x) - 1^{2}}$

Step 3.5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \tan (x)$ and $b = 1$ .

$\sqrt{\sec (x)} = \pm \sqrt{(\tan (x) + 1) (\tan (x) - 1)}$

Step 3.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.6.1

First, use the positive value of the $\pm$ to find the first solution.

$\sqrt{\sec (x)} = \sqrt{(\tan (x) + 1) (\tan (x) - 1)}$

Step 3.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$\sqrt{\sec (x)} = - \sqrt{(\tan (x) + 1) (\tan (x) - 1)}$

Step 3.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$\sqrt{\sec (x)} = \sqrt{(\tan (x) + 1) (\tan (x) - 1)}, - \sqrt{(\tan (x) + 1) (\tan (x) - 1)}$

Step 4

Solve for $x$ in $\sqrt{\sec (x)} = \sqrt{(\tan (x) + 1) (\tan (x) - 1)}$ .

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Step 4.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{\sec (x)}}^{2} = {\sqrt{(\tan (x) + 1) (\tan (x) - 1)}}^{2}$

Step 4.2

Simplify each side of the equation.

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\sec (x)}$ as ${\sec (x)}^{\frac{1}{2}}$ .

${({\sec (x)}^{\frac{1}{2}})}^{2} = {\sqrt{(\tan (x) + 1) (\tan (x) - 1)}}^{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify ${({\sec (x)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1

Multiply the exponents in ${({\sec (x)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${\sec (x)}^{\frac{1}{2} \cdot 2} = {\sqrt{(\tan (x) + 1) (\tan (x) - 1)}}^{2}$

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

${\sec (x)}^{\frac{1}{2} \cdot 2} = {\sqrt{(\tan (x) + 1) (\tan (x) - 1)}}^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

$\sec^{1} (x) = {\sqrt{(\tan (x) + 1) (\tan (x) - 1)}}^{2}$

Step 4.2.2.1.2

Simplify.

$\sec (x) = {\sqrt{(\tan (x) + 1) (\tan (x) - 1)}}^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify ${\sqrt{(\tan (x) + 1) (\tan (x) - 1)}}^{2}$ .

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Step 4.2.3.1.1

Rewrite ${\sqrt{(\tan (x) + 1) (\tan (x) - 1)}}^{2}$ as $(\tan (x) + 1) (\tan (x) - 1)$ .

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Step 4.2.3.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(\tan (x) + 1) (\tan (x) - 1)}$ as ${((\tan (x) + 1) (\tan (x) - 1))}^{\frac{1}{2}}$ .

$\sec (x) = {({((\tan (x) + 1) (\tan (x) - 1))}^{\frac{1}{2}})}^{2}$

Step 4.2.3.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sec (x) = {((\tan (x) + 1) (\tan (x) - 1))}^{\frac{1}{2} \cdot 2}$

Step 4.2.3.1.1.3

Combine $\frac{1}{2}$ and $2$ .

$\sec (x) = {((\tan (x) + 1) (\tan (x) - 1))}^{\frac{2}{2}}$

Step 4.2.3.1.1.4

Cancel the common factor of $2$ .

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Step 4.2.3.1.1.4.1

Cancel the common factor.

$\sec (x) = {((\tan (x) + 1) (\tan (x) - 1))}^{\frac{2}{2}}$

Step 4.2.3.1.1.4.2

Rewrite the expression.

$\sec (x) = {((\tan (x) + 1) (\tan (x) - 1))}^{1}$

Step 4.2.3.1.1.5

Simplify.

$\sec (x) = (\tan (x) + 1) (\tan (x) - 1)$

Step 4.2.3.1.2

Expand $(\tan (x) + 1) (\tan (x) - 1)$ using the FOIL Method.

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Step 4.2.3.1.2.1

Apply the distributive property.

$\sec (x) = \tan (x) (\tan (x) - 1) + 1 (\tan (x) - 1)$

Step 4.2.3.1.2.2

Apply the distributive property.

$\sec (x) = \tan (x) \tan (x) + \tan (x) \cdot - 1 + 1 (\tan (x) - 1)$

Step 4.2.3.1.2.3

Apply the distributive property.

$\sec (x) = \tan (x) \tan (x) + \tan (x) \cdot - 1 + 1 \tan (x) + 1 \cdot - 1$

Step 4.2.3.1.3

Simplify and combine like terms.

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Step 4.2.3.1.3.1

Simplify each term.

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Step 4.2.3.1.3.1.1

Multiply $\tan (x) \tan (x)$ .

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Step 4.2.3.1.3.1.1.1

Raise $\tan (x)$ to the power of $1$ .

$\sec (x) = \tan^{1} (x) \tan (x) + \tan (x) \cdot - 1 + 1 \tan (x) + 1 \cdot - 1$

Step 4.2.3.1.3.1.1.2

Raise $\tan (x)$ to the power of $1$ .

$\sec (x) = \tan^{1} (x) \tan^{1} (x) + \tan (x) \cdot - 1 + 1 \tan (x) + 1 \cdot - 1$

Step 4.2.3.1.3.1.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sec (x) = {\tan (x)}^{1 + 1} + \tan (x) \cdot - 1 + 1 \tan (x) + 1 \cdot - 1$

Step 4.2.3.1.3.1.1.4

Add $1$ and $1$ .

$\sec (x) = \tan^{2} (x) + \tan (x) \cdot - 1 + 1 \tan (x) + 1 \cdot - 1$

Step 4.2.3.1.3.1.2

Move $- 1$ to the left of $\tan (x)$ .

$\sec (x) = \tan^{2} (x) - 1 \cdot \tan (x) + 1 \tan (x) + 1 \cdot - 1$

Step 4.2.3.1.3.1.3

Rewrite $- 1 \tan (x)$ as $- \tan (x)$ .

$\sec (x) = \tan^{2} (x) - \tan (x) + 1 \tan (x) + 1 \cdot - 1$

Step 4.2.3.1.3.1.4

Multiply $\tan (x)$ by $1$ .

$\sec (x) = \tan^{2} (x) - \tan (x) + \tan (x) + 1 \cdot - 1$

Step 4.2.3.1.3.1.5

Multiply $- 1$ by $1$ .

$\sec (x) = \tan^{2} (x) - \tan (x) + \tan (x) - 1$

Step 4.2.3.1.3.2

Add $- \tan (x)$ and $\tan (x)$ .

$\sec (x) = \tan^{2} (x) + 0 - 1$

Step 4.2.3.1.3.3

Add $\tan^{2} (x)$ and $0$ .

$\sec (x) = \tan^{2} (x) - 1$

Step 4.3

Solve for $x$ .

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Step 4.3.1

Subtract $\tan^{2} (x)$ from both sides of the equation.

$\sec (x) - \tan^{2} (x) = - 1$

Step 4.3.2

Replace the $- \tan^{2} (x)$ with $- (\sec^{2} (x) - 1)$ based on the $\tan^{2} (x) + 1 = \sec^{2} (x)$ identity.

$\sec (x) - (\sec^{2} (x) - 1) = - 1$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

$\sec (x) - \sec^{2} (x) + 1 = - 1$

Step 4.3.3.2

Multiply $- 1$ by $- 1$ .

$\sec (x) - \sec^{2} (x) + 1 = - 1$

Step 4.3.4

Reorder the polynomial.

$- \sec^{2} (x) + \sec (x) + 1 = - 1$

Step 4.3.5

Substitute $u$ for $\sec (x)$ .

$- {(u)}^{2} + u + 1 = - 1$

Step 4.3.6

Add $1$ to both sides of the equation.

$- u^{2} + u + 1 + 1 = 0$

Step 4.3.7

Add $1$ and $1$ .

$- u^{2} + u + 2 = 0$

Step 4.3.8

Factor the left side of the equation.

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Step 4.3.8.1

Factor $- 1$ out of $- u^{2} + u + 2$ .

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Step 4.3.8.1.1

Factor $- 1$ out of $- u^{2}$ .

$- u^{2} + u + 2 = 0$

Step 4.3.8.1.2

Factor $- 1$ out of $u$ .

$- u^{2} - 1 (- u) + 2 = 0$

Step 4.3.8.1.3

Rewrite $2$ as $- 1 (- 2)$ .

$- u^{2} - 1 (- u) - 1 \cdot - 2 = 0$

Step 4.3.8.1.4

Factor $- 1$ out of $- u^{2} - 1 (- u)$ .

$- (u^{2} - u) - 1 \cdot - 2 = 0$

Step 4.3.8.1.5

Factor $- 1$ out of $- (u^{2} - u) - 1 (- 2)$ .

$- (u^{2} - u - 2) = 0$

Step 4.3.8.2

Factor.

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Step 4.3.8.2.1

Factor $u^{2} - u - 2$ using the AC method.

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Step 4.3.8.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 4.3.8.2.1.2

Write the factored form using these integers.

$- ((u - 2) (u + 1)) = 0$

Step 4.3.8.2.2

Remove unnecessary parentheses.

$- (u - 2) (u + 1) = 0$

Step 4.3.9

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

$u - 2 = 0$

$u + 1 = 0$

Step 4.3.10

Set $u - 2$ equal to $0$ and solve for $u$ .

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Step 4.3.10.1

Set $u - 2$ equal to $0$ .

$u - 2 = 0$

Step 4.3.10.2

Add $2$ to both sides of the equation.

$u = 2$

Step 4.3.11

Set $u + 1$ equal to $0$ and solve for $u$ .

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Step 4.3.11.1

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 4.3.11.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 4.3.12

The final solution is all the values that make $- (u - 2) (u + 1) = 0$ true.

$u = 2, - 1$

Step 4.3.13

Substitute $\sec (x)$ for $u$ .

$\sec (x) = 2, - 1$

Step 4.3.14

Set up each of the solutions to solve for $x$ .

$\sec (x) = 2$

$\sec (x) = - 1$

Step 4.3.15

Solve for $x$ in $\sec (x) = 2$ .

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Step 4.3.15.1

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

$x = arcsec (2)$

Step 4.3.15.2

Simplify the right side.

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Step 4.3.15.2.1

The exact value of $arcsec (2)$ is $\frac{π}{3}$ .

$x = \frac{π}{3}$

Step 4.3.15.3

The secant 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.

$x = 2 π - \frac{π}{3}$

Step 4.3.15.4

Simplify $2 π - \frac{π}{3}$ .

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Step 4.3.15.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = 2 π \cdot \frac{3}{3} - \frac{π}{3}$

Step 4.3.15.4.2

Combine fractions.

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Step 4.3.15.4.2.1

Combine $2 π$ and $\frac{3}{3}$ .

$x = \frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 4.3.15.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - π}{3}$

Step 4.3.15.4.3

Simplify the numerator.

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Step 4.3.15.4.3.1

Multiply $3$ by $2$ .

$x = \frac{6 π - π}{3}$

Step 4.3.15.4.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{3}$

Step 4.3.15.5

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

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Step 4.3.15.5.1

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

$\frac{2 π}{| b |}$

Step 4.3.15.5.2

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

$\frac{2 π}{| 1 |}$

Step 4.3.15.5.3

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

$\frac{2 π}{1}$

Step 4.3.15.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.3.15.6

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

$x = \frac{π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer $n$

Step 4.3.16

Solve for $x$ in $\sec (x) = - 1$ .

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Step 4.3.16.1

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

$x = arcsec (- 1)$

Step 4.3.16.2

Simplify the right side.

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Step 4.3.16.2.1

The exact value of $arcsec (- 1)$ is $π$ .

$x = π$

Step 4.3.16.3

The secant function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 π - π$

Step 4.3.16.4

Subtract $π$ from $2 π$ .

$x = π$

Step 4.3.16.5

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

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Step 4.3.16.5.1

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

$\frac{2 π}{| b |}$

Step 4.3.16.5.2

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

$\frac{2 π}{| 1 |}$

Step 4.3.16.5.3

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

$\frac{2 π}{1}$

Step 4.3.16.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.3.16.6

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

$x = π + 2 π n$ , for any integer $n$

Step 4.3.17

List all of the solutions.

$x = \frac{π}{3} + 2 π n, \frac{5 π}{3} + 2 π n, π + 2 π n$ , for any integer $n$

Step 4.3.18

Consolidate the answers.

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

Step 5

Solve for $x$ in $\sqrt{\sec (x)} = - \sqrt{(\tan (x) + 1) (\tan (x) - 1)}$ .

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Step 5.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{\sec (x)}}^{2} = {(- \sqrt{(\tan (x) + 1) (\tan (x) - 1)})}^{2}$

Step 5.2

Simplify each side of the equation.

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Step 5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\sec (x)}$ as ${\sec (x)}^{\frac{1}{2}}$ .

${({\sec (x)}^{\frac{1}{2}})}^{2} = {(- \sqrt{(\tan (x) + 1) (\tan (x) - 1)})}^{2}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Simplify ${({\sec (x)}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.2.1.1

Multiply the exponents in ${({\sec (x)}^{\frac{1}{2}})}^{2}$ .

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Step 5.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${\sec (x)}^{\frac{1}{2} \cdot 2} = {(- \sqrt{(\tan (x) + 1) (\tan (x) - 1)})}^{2}$

Step 5.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.2.1.1.2.1

Cancel the common factor.

${\sec (x)}^{\frac{1}{2} \cdot 2} = {(- \sqrt{(\tan (x) + 1) (\tan (x) - 1)})}^{2}$

Step 5.2.2.1.1.2.2

Rewrite the expression.

$\sec^{1} (x) = {(- \sqrt{(\tan (x) + 1) (\tan (x) - 1)})}^{2}$

Step 5.2.2.1.2

Simplify.

$\sec (x) = {(- \sqrt{(\tan (x) + 1) (\tan (x) - 1)})}^{2}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify ${(- \sqrt{(\tan (x) + 1) (\tan (x) - 1)})}^{2}$ .

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Step 5.2.3.1.1

Simplify by cancelling exponent with radical.

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Step 5.2.3.1.1.1

Apply the product rule to $- \sqrt{(\tan (x) + 1) (\tan (x) - 1)}$ .

$\sec (x) = {(- 1)}^{2} {\sqrt{(\tan (x) + 1) (\tan (x) - 1)}}^{2}$

Step 5.2.3.1.1.2

Simplify the expression.

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Step 5.2.3.1.1.2.1

Raise $- 1$ to the power of $2$ .

$\sec (x) = 1 {\sqrt{(\tan (x) + 1) (\tan (x) - 1)}}^{2}$

Step 5.2.3.1.1.2.2

Multiply ${\sqrt{(\tan (x) + 1) (\tan (x) - 1)}}^{2}$ by $1$ .

$\sec (x) = {\sqrt{(\tan (x) + 1) (\tan (x) - 1)}}^{2}$

Step 5.2.3.1.1.3

Rewrite ${\sqrt{(\tan (x) + 1) (\tan (x) - 1)}}^{2}$ as $(\tan (x) + 1) (\tan (x) - 1)$ .

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Step 5.2.3.1.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(\tan (x) + 1) (\tan (x) - 1)}$ as ${((\tan (x) + 1) (\tan (x) - 1))}^{\frac{1}{2}}$ .

$\sec (x) = {({((\tan (x) + 1) (\tan (x) - 1))}^{\frac{1}{2}})}^{2}$

Step 5.2.3.1.1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sec (x) = {((\tan (x) + 1) (\tan (x) - 1))}^{\frac{1}{2} \cdot 2}$

Step 5.2.3.1.1.3.3

Combine $\frac{1}{2}$ and $2$ .

$\sec (x) = {((\tan (x) + 1) (\tan (x) - 1))}^{\frac{2}{2}}$

Step 5.2.3.1.1.3.4

Cancel the common factor of $2$ .

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Step 5.2.3.1.1.3.4.1

Cancel the common factor.

$\sec (x) = {((\tan (x) + 1) (\tan (x) - 1))}^{\frac{2}{2}}$

Step 5.2.3.1.1.3.4.2

Rewrite the expression.

$\sec (x) = {((\tan (x) + 1) (\tan (x) - 1))}^{1}$

Step 5.2.3.1.1.3.5

Simplify.

$\sec (x) = (\tan (x) + 1) (\tan (x) - 1)$

Step 5.2.3.1.2

Expand $(\tan (x) + 1) (\tan (x) - 1)$ using the FOIL Method.

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Step 5.2.3.1.2.1

Apply the distributive property.

$\sec (x) = \tan (x) (\tan (x) - 1) + 1 (\tan (x) - 1)$

Step 5.2.3.1.2.2

Apply the distributive property.

$\sec (x) = \tan (x) \tan (x) + \tan (x) \cdot - 1 + 1 (\tan (x) - 1)$

Step 5.2.3.1.2.3

Apply the distributive property.

$\sec (x) = \tan (x) \tan (x) + \tan (x) \cdot - 1 + 1 \tan (x) + 1 \cdot - 1$

Step 5.2.3.1.3

Simplify and combine like terms.

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Step 5.2.3.1.3.1

Simplify each term.

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Step 5.2.3.1.3.1.1

Multiply $\tan (x) \tan (x)$ .

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Step 5.2.3.1.3.1.1.1

Raise $\tan (x)$ to the power of $1$ .

$\sec (x) = \tan^{1} (x) \tan (x) + \tan (x) \cdot - 1 + 1 \tan (x) + 1 \cdot - 1$

Step 5.2.3.1.3.1.1.2

Raise $\tan (x)$ to the power of $1$ .

$\sec (x) = \tan^{1} (x) \tan^{1} (x) + \tan (x) \cdot - 1 + 1 \tan (x) + 1 \cdot - 1$

Step 5.2.3.1.3.1.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sec (x) = {\tan (x)}^{1 + 1} + \tan (x) \cdot - 1 + 1 \tan (x) + 1 \cdot - 1$

Step 5.2.3.1.3.1.1.4

Add $1$ and $1$ .

$\sec (x) = \tan^{2} (x) + \tan (x) \cdot - 1 + 1 \tan (x) + 1 \cdot - 1$

Step 5.2.3.1.3.1.2

Move $- 1$ to the left of $\tan (x)$ .

$\sec (x) = \tan^{2} (x) - 1 \cdot \tan (x) + 1 \tan (x) + 1 \cdot - 1$

Step 5.2.3.1.3.1.3

Rewrite $- 1 \tan (x)$ as $- \tan (x)$ .

$\sec (x) = \tan^{2} (x) - \tan (x) + 1 \tan (x) + 1 \cdot - 1$

Step 5.2.3.1.3.1.4

Multiply $\tan (x)$ by $1$ .

$\sec (x) = \tan^{2} (x) - \tan (x) + \tan (x) + 1 \cdot - 1$

Step 5.2.3.1.3.1.5

Multiply $- 1$ by $1$ .

$\sec (x) = \tan^{2} (x) - \tan (x) + \tan (x) - 1$

Step 5.2.3.1.3.2

Add $- \tan (x)$ and $\tan (x)$ .

$\sec (x) = \tan^{2} (x) + 0 - 1$

Step 5.2.3.1.3.3

Add $\tan^{2} (x)$ and $0$ .

$\sec (x) = \tan^{2} (x) - 1$

Step 5.3

Solve for $x$ .

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Step 5.3.1

Subtract $\tan^{2} (x)$ from both sides of the equation.

$\sec (x) - \tan^{2} (x) = - 1$

Step 5.3.2

Replace the $- \tan^{2} (x)$ with $- (\sec^{2} (x) - 1)$ based on the $\tan^{2} (x) + 1 = \sec^{2} (x)$ identity.

$\sec (x) - (\sec^{2} (x) - 1) = - 1$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$\sec (x) - \sec^{2} (x) + 1 = - 1$

Step 5.3.3.2

Multiply $- 1$ by $- 1$ .

$\sec (x) - \sec^{2} (x) + 1 = - 1$

Step 5.3.4

Reorder the polynomial.

$- \sec^{2} (x) + \sec (x) + 1 = - 1$

Step 5.3.5

Substitute $u$ for $\sec (x)$ .

$- {(u)}^{2} + u + 1 = - 1$

Step 5.3.6

Add $1$ to both sides of the equation.

$- u^{2} + u + 1 + 1 = 0$

Step 5.3.7

Add $1$ and $1$ .

$- u^{2} + u + 2 = 0$

Step 5.3.8

Factor the left side of the equation.

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Step 5.3.8.1

Factor $- 1$ out of $- u^{2} + u + 2$ .

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Step 5.3.8.1.1

Factor $- 1$ out of $- u^{2}$ .

$- u^{2} + u + 2 = 0$

Step 5.3.8.1.2

Factor $- 1$ out of $u$ .

$- u^{2} - 1 (- u) + 2 = 0$

Step 5.3.8.1.3

Rewrite $2$ as $- 1 (- 2)$ .

$- u^{2} - 1 (- u) - 1 \cdot - 2 = 0$

Step 5.3.8.1.4

Factor $- 1$ out of $- u^{2} - 1 (- u)$ .

$- (u^{2} - u) - 1 \cdot - 2 = 0$

Step 5.3.8.1.5

Factor $- 1$ out of $- (u^{2} - u) - 1 (- 2)$ .

$- (u^{2} - u - 2) = 0$

Step 5.3.8.2

Factor.

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Step 5.3.8.2.1

Factor $u^{2} - u - 2$ using the AC method.

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Step 5.3.8.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 5.3.8.2.1.2

Write the factored form using these integers.

$- ((u - 2) (u + 1)) = 0$

Step 5.3.8.2.2

Remove unnecessary parentheses.

$- (u - 2) (u + 1) = 0$

Step 5.3.9

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

$u - 2 = 0$

$u + 1 = 0$

Step 5.3.10

Set $u - 2$ equal to $0$ and solve for $u$ .

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Step 5.3.10.1

Set $u - 2$ equal to $0$ .

$u - 2 = 0$

Step 5.3.10.2

Add $2$ to both sides of the equation.

$u = 2$

Step 5.3.11

Set $u + 1$ equal to $0$ and solve for $u$ .

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Step 5.3.11.1

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 5.3.11.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 5.3.12

The final solution is all the values that make $- (u - 2) (u + 1) = 0$ true.

$u = 2, - 1$

Step 5.3.13

Substitute $\sec (x)$ for $u$ .

$\sec (x) = 2, - 1$

Step 5.3.14

Set up each of the solutions to solve for $x$ .

$\sec (x) = 2$

$\sec (x) = - 1$

Step 5.3.15

Solve for $x$ in $\sec (x) = 2$ .

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Step 5.3.15.1

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

$x = arcsec (2)$

Step 5.3.15.2

Simplify the right side.

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Step 5.3.15.2.1

The exact value of $arcsec (2)$ is $\frac{π}{3}$ .

$x = \frac{π}{3}$

Step 5.3.15.3

The secant 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.

$x = 2 π - \frac{π}{3}$

Step 5.3.15.4

Simplify $2 π - \frac{π}{3}$ .

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Step 5.3.15.4.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = 2 π \cdot \frac{3}{3} - \frac{π}{3}$

Step 5.3.15.4.2

Combine fractions.

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Step 5.3.15.4.2.1

Combine $2 π$ and $\frac{3}{3}$ .

$x = \frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 5.3.15.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - π}{3}$

Step 5.3.15.4.3

Simplify the numerator.

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Step 5.3.15.4.3.1

Multiply $3$ by $2$ .

$x = \frac{6 π - π}{3}$

Step 5.3.15.4.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{3}$

Step 5.3.15.5

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

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Step 5.3.15.5.1

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

$\frac{2 π}{| b |}$

Step 5.3.15.5.2

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

$\frac{2 π}{| 1 |}$

Step 5.3.15.5.3

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

$\frac{2 π}{1}$

Step 5.3.15.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.3.15.6

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

$x = \frac{π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer $n$

Step 5.3.16

Solve for $x$ in $\sec (x) = - 1$ .

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Step 5.3.16.1

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

$x = arcsec (- 1)$

Step 5.3.16.2

Simplify the right side.

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Step 5.3.16.2.1

The exact value of $arcsec (- 1)$ is $π$ .

$x = π$

Step 5.3.16.3

The secant function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 π - π$

Step 5.3.16.4

Subtract $π$ from $2 π$ .

$x = π$

Step 5.3.16.5

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

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Step 5.3.16.5.1

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

$\frac{2 π}{| b |}$

Step 5.3.16.5.2

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

$\frac{2 π}{| 1 |}$

Step 5.3.16.5.3

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

$\frac{2 π}{1}$

Step 5.3.16.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.3.16.6

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

$x = π + 2 π n$ , for any integer $n$

Step 5.3.17

List all of the solutions.

$x = \frac{π}{3} + 2 π n, \frac{5 π}{3} + 2 π n, π + 2 π n$ , for any integer $n$

Step 5.3.18

Consolidate the answers.

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

Step 6

List all of the solutions.

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

Step 7

Exclude the solutions that do not make $(\tan (x) + \sqrt{\sec (x)}) (\tan (x) - \sqrt{\sec (x)}) = 1$ true.

$x = \frac{π}{3} + 2 π n, \frac{π}{3} + 2 π n, \frac{5 π}{3} + 2 π n, \frac{5 π}{3} + 2 π n$ , for any integer $n$