Precalculus Examples

$\tan^{2} (x) \cos (x) - \cos (x) = 0$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

Cancel the common factor.

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

Step 1.1.3.3

Rewrite the expression.

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

Step 2

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) (\frac{\sin^{2} (x)}{\cos (x)} - \cos (x)) = \cos (x) \cdot 0$

Step 3

Apply the distributive property.

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

$\sin^{2} (x) + \cos (x) (- \cos (x)) = \cos (x) \cdot 0$

Step 5

Rewrite using the commutative property of multiplication.

$\sin^{2} (x) - \cos (x) \cos (x) = \cos (x) \cdot 0$

Step 6

Multiply $- \cos (x) \cos (x)$ .

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Step 6.1

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

$\sin^{2} (x) - (\cos^{1} (x) \cos (x)) = \cos (x) \cdot 0$

Step 6.2

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

$\sin^{2} (x) - (\cos^{1} (x) \cos^{1} (x)) = \cos (x) \cdot 0$

Step 6.3

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

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

Step 6.4

Add $1$ and $1$ .

$\sin^{2} (x) - \cos^{2} (x) = \cos (x) \cdot 0$

Step 7

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

$\sin^{2} (x) - \cos^{2} (x) = 0$

Step 8

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

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

Step 9

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

$\sin (x) + \cos (x) = 0$

$\sin (x) - \cos (x) = 0$

Step 10

Set $\sin (x) + \cos (x)$ equal to $0$ and solve for $x$ .

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Step 10.1

Set $\sin (x) + \cos (x)$ equal to $0$ .

$\sin (x) + \cos (x) = 0$

Step 10.2

Solve $\sin (x) + \cos (x) = 0$ for $x$ .

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Step 10.2.1

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

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

Step 10.2.2

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

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

Step 10.2.3

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

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Step 10.2.3.1

Cancel the common factor.

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

Step 10.2.3.2

Rewrite the expression.

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

Step 10.2.4

Separate fractions.

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

Step 10.2.5

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

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

Step 10.2.6

Divide $0$ by $1$ .

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

Step 10.2.7

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

$\tan (x) + 1 = 0$

Step 10.2.8

Subtract $1$ from both sides of the equation.

$\tan (x) = - 1$

Step 10.2.9

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

$x = \arctan (- 1)$

Step 10.2.10

Simplify the right side.

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Step 10.2.10.1

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

$x = - \frac{π}{4}$

Step 10.2.11

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 10.2.12

Simplify the expression to find the second solution.

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Step 10.2.12.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$x = - \frac{π}{4} - π + 2 π$

Step 10.2.12.2

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

$x = \frac{3 π}{4}$

Step 10.2.13

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

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Step 10.2.13.1

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

$\frac{π}{| b |}$

Step 10.2.13.2

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

$\frac{π}{| 1 |}$

Step 10.2.13.3

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

$\frac{π}{1}$

Step 10.2.13.4

Divide $π$ by $1$ .

$π$

Step 10.2.14

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

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Step 10.2.14.1

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

$- \frac{π}{4} + π$

Step 10.2.14.2

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

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 10.2.14.3

Combine fractions.

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Step 10.2.14.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 10.2.14.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 10.2.14.4

Simplify the numerator.

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Step 10.2.14.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 10.2.14.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 10.2.14.5

List the new angles.

$x = \frac{3 π}{4}$

Step 10.2.15

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 11

Set $\sin (x) - \cos (x)$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $\sin (x) - \cos (x)$ equal to $0$ .

$\sin (x) - \cos (x) = 0$

Step 11.2

Solve $\sin (x) - \cos (x) = 0$ for $x$ .

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Step 11.2.1

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

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

Step 11.2.2

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

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

Step 11.2.3

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

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Step 11.2.3.1

Cancel the common factor.

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

Step 11.2.3.2

Divide $- 1$ by $1$ .

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

Step 11.2.4

Separate fractions.

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

Step 11.2.5

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

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

Step 11.2.6

Divide $0$ by $1$ .

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

Step 11.2.7

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

$\tan (x) - 1 = 0$

Step 11.2.8

Add $1$ to both sides of the equation.

$\tan (x) = 1$

Step 11.2.9

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

$x = \arctan (1)$

Step 11.2.10

Simplify the right side.

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Step 11.2.10.1

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

$x = \frac{π}{4}$

Step 11.2.11

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.

$x = π + \frac{π}{4}$

Step 11.2.12

Simplify $π + \frac{π}{4}$ .

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Step 11.2.12.1

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

$x = π \cdot \frac{4}{4} + \frac{π}{4}$

Step 11.2.12.2

Combine fractions.

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Step 11.2.12.2.1

Combine $π$ and $\frac{4}{4}$ .

$x = \frac{π \cdot 4}{4} + \frac{π}{4}$

Step 11.2.12.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Step 11.2.12.3

Simplify the numerator.

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Step 11.2.12.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{4}$

Step 11.2.12.3.2

Add $4 π$ and $π$ .

$x = \frac{5 π}{4}$

Step 11.2.13

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

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Step 11.2.13.1

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

$\frac{π}{| b |}$

Step 11.2.13.2

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

$\frac{π}{| 1 |}$

Step 11.2.13.3

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

$\frac{π}{1}$

Step 11.2.13.4

Divide $π$ by $1$ .

$π$

Step 11.2.14

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

$x = \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

Step 12

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

$x = \frac{3 π}{4} + π n, \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

Step 13

Consolidate the answers.

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