Precalculus Examples

$(\tan (x) + 1) (\cos (x) - 1) = 0$

Step 1

Simplify the left side.

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Step 1.1

Simplify $(\tan (x) + 1) (\cos (x) - 1)$ .

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Step 1.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$(\frac{\sin (x)}{\cos (x)} + 1) (\cos (x) - 1) = 0$

Step 1.1.2

Expand $(\frac{\sin (x)}{\cos (x)} + 1) (\cos (x) - 1)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$\frac{\sin (x)}{\cos (x)} (\cos (x) - 1) + 1 (\cos (x) - 1) = 0$

Step 1.1.2.2

Apply the distributive property.

$\frac{\sin (x)}{\cos (x)} \cos (x) + \frac{\sin (x)}{\cos (x)} \cdot - 1 + 1 (\cos (x) - 1) = 0$

Step 1.1.2.3

Apply the distributive property.

$\frac{\sin (x)}{\cos (x)} \cos (x) + \frac{\sin (x)}{\cos (x)} \cdot - 1 + 1 \cos (x) + 1 \cdot - 1 = 0$

Step 1.1.3

Simplify each term.

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Step 1.1.3.1

Cancel the common factor of $\cos (x)$ .

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Step 1.1.3.1.1

Cancel the common factor.

$\frac{\sin (x)}{\cos (x)} \cos (x) + \frac{\sin (x)}{\cos (x)} \cdot - 1 + 1 \cos (x) + 1 \cdot - 1 = 0$

Step 1.1.3.1.2

Rewrite the expression.

$\sin (x) + \frac{\sin (x)}{\cos (x)} \cdot - 1 + 1 \cos (x) + 1 \cdot - 1 = 0$

Step 1.1.3.2

Combine $\frac{\sin (x)}{\cos (x)}$ and $- 1$ .

$\sin (x) + \frac{\sin (x) \cdot - 1}{\cos (x)} + 1 \cos (x) + 1 \cdot - 1 = 0$

Step 1.1.3.3

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

$\sin (x) + \frac{- 1 \cdot \sin (x)}{\cos (x)} + 1 \cos (x) + 1 \cdot - 1 = 0$

Step 1.1.3.4

Move the negative in front of the fraction.

$\sin (x) - \frac{\sin (x)}{\cos (x)} + 1 \cos (x) + 1 \cdot - 1 = 0$

Step 1.1.3.5

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

$\sin (x) - \frac{\sin (x)}{\cos (x)} + \cos (x) + 1 \cdot - 1 = 0$

Step 1.1.3.6

Multiply $- 1$ by $1$ .

$\sin (x) - \frac{\sin (x)}{\cos (x)} + \cos (x) - 1 = 0$

Step 1.1.4

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\sin (x) - \tan (x) + \cos (x) - 1 = 0$

Step 2

Divide each term in the equation by $\cos (x)$ .

$\frac{\sin (x)}{\cos (x)} + \frac{- \tan (x)}{\cos (x)} + \frac{\cos (x)}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\tan (x) + \frac{- \tan (x)}{\cos (x)} + \frac{\cos (x)}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 4

Separate fractions.

$\tan (x) + \frac{- 1}{1} \cdot \frac{\tan (x)}{\cos (x)} + \frac{\cos (x)}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 5

Rewrite $\tan (x)$ in terms of sines and cosines.

$\tan (x) + \frac{- 1}{1} \cdot \frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)} + \frac{\cos (x)}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 6

Rewrite $\frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)}$ as a product.

$\tan (x) + \frac{- 1}{1} \cdot (\frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)}) + \frac{\cos (x)}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 7

Combine fractions.

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Step 7.1

Multiply $\frac{\sin (x)}{\cos (x)}$ by $\frac{1}{\cos (x)}$ .

$\tan (x) + \frac{- 1}{1} \cdot \frac{\sin (x)}{\cos (x) \cos (x)} + \frac{\cos (x)}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 7.2

Divide $- 1$ by $1$ .

$\tan (x) - \frac{\sin (x)}{\cos (x) \cos (x)} + \frac{\cos (x)}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 8

Simplify the denominator.

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Step 8.1

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

$\tan (x) - \frac{\sin (x)}{{\cos (x)}^{1 + 1}} + \frac{\cos (x)}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 8.2

Add $1$ and $1$ .

$\tan (x) - \frac{\sin (x)}{\cos^{2} (x)} + \frac{\cos (x)}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 9

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\tan (x) - \frac{\sin (x)}{\cos (x) \cos (x)} + \frac{\cos (x)}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 10

Separate fractions.

$\tan (x) - (\frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}) + \frac{\cos (x)}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 11

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\tan (x) - (\frac{1}{\cos (x)} \cdot \tan (x)) + \frac{\cos (x)}{\cos (x)} + \frac{- 1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 12

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 13

Cancel the common factor of $\cos (x)$ .

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Step 13.1

Cancel the common factor.

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

Step 13.2

Rewrite the expression.

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

Step 14

Separate fractions.

$\tan (x) - \sec (x) \tan (x) + 1 + \frac{- 1}{1} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 15

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\tan (x) - \sec (x) \tan (x) + 1 + \frac{- 1}{1} \cdot \sec (x) = \frac{0}{\cos (x)}$

Step 16

Divide $- 1$ by $1$ .

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

Step 17

Separate fractions.

$\tan (x) - \sec (x) \tan (x) + 1 - \sec (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 18

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 19

Divide $0$ by $1$ .

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

Step 20

Multiply $0$ by $\sec (x)$ .

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

Step 21

Factor $\tan (x) - \sec (x) \tan (x) + 1 - \sec (x)$ .

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Step 21.1

Factor out the greatest common factor from each group.

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Step 21.1.1

Group the first two terms and the last two terms.

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

Step 21.1.2

Factor out the greatest common factor (GCF) from each group.

$\tan (x) \cdot (1 - \sec (x)) + 1 (1 - \sec (x)) = 0$

Step 21.2

Factor the polynomial by factoring out the greatest common factor, $1 - \sec (x)$ .

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

Step 22

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

$1 - \sec (x) = 0$

$\tan (x) + 1 = 0$

Step 23

Set $1 - \sec (x)$ equal to $0$ and solve for $x$ .

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Step 23.1

Set $1 - \sec (x)$ equal to $0$ .

$1 - \sec (x) = 0$

Step 23.2

Solve $1 - \sec (x) = 0$ for $x$ .

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Step 23.2.1

Subtract $1$ from both sides of the equation.

$- \sec (x) = - 1$

Step 23.2.2

Divide each term in $- \sec (x) = - 1$ by $- 1$ and simplify.

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Step 23.2.2.1

Divide each term in $- \sec (x) = - 1$ by $- 1$ .

$\frac{- \sec (x)}{- 1} = \frac{- 1}{- 1}$

Step 23.2.2.2

Simplify the left side.

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Step 23.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{\sec (x)}{1} = \frac{- 1}{- 1}$

Step 23.2.2.2.2

Divide $\sec (x)$ by $1$ .

$\sec (x) = \frac{- 1}{- 1}$

Step 23.2.2.3

Simplify the right side.

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Step 23.2.2.3.1

Divide $- 1$ by $- 1$ .

$\sec (x) = 1$

Step 23.2.3

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

$x = arcsec (1)$

Step 23.2.4

Simplify the right side.

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Step 23.2.4.1

The exact value of $arcsec (1)$ is $0$ .

$x = 0$

Step 23.2.5

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.

$x = 2 π - 0$

Step 23.2.6

Subtract $0$ from $2 π$ .

$x = 2 π$

Step 23.2.7

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

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Step 23.2.7.1

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

$\frac{2 π}{| b |}$

Step 23.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 23.2.7.3

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

$\frac{2 π}{1}$

Step 23.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 23.2.8

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

$x = 2 π n, 2 π + 2 π n$ , for any integer $n$

Step 24

Set $\tan (x) + 1$ equal to $0$ and solve for $x$ .

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Step 24.1

Set $\tan (x) + 1$ equal to $0$ .

$\tan (x) + 1 = 0$

Step 24.2

Solve $\tan (x) + 1 = 0$ for $x$ .

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Step 24.2.1

Subtract $1$ from both sides of the equation.

$\tan (x) = - 1$

Step 24.2.2

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

$x = \arctan (- 1)$

Step 24.2.3

Simplify the right side.

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Step 24.2.3.1

The exact value of $\arctan (- 1)$ is $- \frac{π}{4}$ .

$x = - \frac{π}{4}$

Step 24.2.4

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

$x = - \frac{π}{4} - π$

Step 24.2.5

Simplify the expression to find the second solution.

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Step 24.2.5.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$x = - \frac{π}{4} - π + 2 π$

Step 24.2.5.2

The resulting angle of $\frac{3 π}{4}$ is positive and coterminal with $- \frac{π}{4} - π$ .

$x = \frac{3 π}{4}$

Step 24.2.6

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

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Step 24.2.6.1

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

$\frac{π}{| b |}$

Step 24.2.6.2

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

$\frac{π}{| 1 |}$

Step 24.2.6.3

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

$\frac{π}{1}$

Step 24.2.6.4

Divide $π$ by $1$ .

$π$

Step 24.2.7

Add $π$ to every negative angle to get positive angles.

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Step 24.2.7.1

Add $π$ to $- \frac{π}{4}$ to find the positive angle.

$- \frac{π}{4} + π$

Step 24.2.7.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 24.2.7.3

Combine fractions.

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Step 24.2.7.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 24.2.7.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 24.2.7.4

Simplify the numerator.

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Step 24.2.7.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 24.2.7.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 24.2.7.5

List the new angles.

$x = \frac{3 π}{4}$

Step 24.2.8

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

$x = \frac{3 π}{4} + π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 25

The final solution is all the values that make $(1 - \sec (x)) (\tan (x) + 1) = 0$ true.

$x = 2 π n, 2 π + 2 π n, \frac{3 π}{4} + π n$ , for any integer $n$

Step 26

Consolidate $2 π n$ and $2 π + 2 π n$ to $2 π n$ .

$x = 2 π n, \frac{3 π}{4} + π n$ , for any integer $n$