Precalculus Examples

$\sec (θ) \tan (θ) - \cos (θ) \cot (θ) = \sin (θ)$

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 $\sec (θ)$ in terms of sines and cosines.

$\frac{1}{\cos (θ)} \tan (θ) - \cos (θ) \cot (θ) = \sin (θ)$

Step 1.1.2

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

$\frac{1}{\cos (θ)} \cdot \frac{\sin (θ)}{\cos (θ)} - \cos (θ) \cot (θ) = \sin (θ)$

Step 1.1.3

Multiply $\frac{1}{\cos (θ)} \cdot \frac{\sin (θ)}{\cos (θ)}$ .

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Step 1.1.3.1

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

$\frac{\sin (θ)}{\cos (θ) \cos (θ)} - \cos (θ) \cot (θ) = \sin (θ)$

Step 1.1.3.2

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

$\frac{\sin (θ)}{\cos^{1} (θ) \cos (θ)} - \cos (θ) \cot (θ) = \sin (θ)$

Step 1.1.3.3

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

$\frac{\sin (θ)}{\cos^{1} (θ) \cos^{1} (θ)} - \cos (θ) \cot (θ) = \sin (θ)$

Step 1.1.3.4

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

$\frac{\sin (θ)}{{\cos (θ)}^{1 + 1}} - \cos (θ) \cot (θ) = \sin (θ)$

Step 1.1.3.5

Add $1$ and $1$ .

$\frac{\sin (θ)}{\cos^{2} (θ)} - \cos (θ) \cot (θ) = \sin (θ)$

Step 1.1.4

Rewrite $\cot (θ)$ in terms of sines and cosines.

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

Step 1.1.5

Multiply $- \cos (θ) \frac{\cos (θ)}{\sin (θ)}$ .

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Step 1.1.5.1

Combine $\frac{\cos (θ)}{\sin (θ)}$ and $\cos (θ)$ .

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

Step 1.1.5.2

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

$\frac{\sin (θ)}{\cos^{2} (θ)} - \frac{\cos^{1} (θ) \cos (θ)}{\sin (θ)} = \sin (θ)$

Step 1.1.5.3

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

$\frac{\sin (θ)}{\cos^{2} (θ)} - \frac{\cos^{1} (θ) \cos^{1} (θ)}{\sin (θ)} = \sin (θ)$

Step 1.1.5.4

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

$\frac{\sin (θ)}{\cos^{2} (θ)} - \frac{{\cos (θ)}^{1 + 1}}{\sin (θ)} = \sin (θ)$

Step 1.1.5.5

Add $1$ and $1$ .

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

Step 2

Multiply both sides of the equation by $\sin (θ)$ .

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

Step 3

Apply the distributive property.

$\sin (θ) \frac{\sin (θ)}{\cos^{2} (θ)} + \sin (θ) (- \frac{\cos^{2} (θ)}{\sin (θ)}) = \sin (θ) \sin (θ)$

Step 4

Multiply $\sin (θ) \frac{\sin (θ)}{\cos^{2} (θ)}$ .

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Step 4.1

Combine $\sin (θ)$ and $\frac{\sin (θ)}{\cos^{2} (θ)}$ .

$\frac{\sin (θ) \sin (θ)}{\cos^{2} (θ)} + \sin (θ) (- \frac{\cos^{2} (θ)}{\sin (θ)}) = \sin (θ) \sin (θ)$

Step 4.2

Raise $\sin (θ)$ to the power of $1$ .

$\frac{\sin^{1} (θ) \sin (θ)}{\cos^{2} (θ)} + \sin (θ) (- \frac{\cos^{2} (θ)}{\sin (θ)}) = \sin (θ) \sin (θ)$

Step 4.3

Raise $\sin (θ)$ to the power of $1$ .

$\frac{\sin^{1} (θ) \sin^{1} (θ)}{\cos^{2} (θ)} + \sin (θ) (- \frac{\cos^{2} (θ)}{\sin (θ)}) = \sin (θ) \sin (θ)$

Step 4.4

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

$\frac{{\sin (θ)}^{1 + 1}}{\cos^{2} (θ)} + \sin (θ) (- \frac{\cos^{2} (θ)}{\sin (θ)}) = \sin (θ) \sin (θ)$

Step 4.5

Add $1$ and $1$ .

$\frac{\sin^{2} (θ)}{\cos^{2} (θ)} + \sin (θ) (- \frac{\cos^{2} (θ)}{\sin (θ)}) = \sin (θ) \sin (θ)$

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

Cancel the common factor of $\sin (θ)$ .

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Step 6.1

Factor $\sin (θ)$ out of $- \sin (θ)$ .

$\frac{\sin^{2} (θ)}{\cos^{2} (θ)} + \sin (θ) \cdot - 1 \frac{\cos^{2} (θ)}{\sin (θ)} = \sin (θ) \sin (θ)$

Step 6.2

Cancel the common factor.

$\frac{\sin^{2} (θ)}{\cos^{2} (θ)} + \sin (θ) \cdot - 1 \frac{\cos^{2} (θ)}{\sin (θ)} = \sin (θ) \sin (θ)$

Step 6.3

Rewrite the expression.

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

Step 7

Multiply $\sin (θ) \sin (θ)$ .

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Step 7.1

Raise $\sin (θ)$ to the power of $1$ .

$\frac{\sin^{2} (θ)}{\cos^{2} (θ)} - \cos^{2} (θ) = \sin^{1} (θ) \sin (θ)$

Step 7.2

Raise $\sin (θ)$ to the power of $1$ .

$\frac{\sin^{2} (θ)}{\cos^{2} (θ)} - \cos^{2} (θ) = \sin^{1} (θ) \sin^{1} (θ)$

Step 7.3

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

$\frac{\sin^{2} (θ)}{\cos^{2} (θ)} - \cos^{2} (θ) = {\sin (θ)}^{1 + 1}$

Step 7.4

Add $1$ and $1$ .

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

Step 8

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

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

Step 9

Simplify $\frac{\sin^{2} (θ)}{\cos^{2} (θ)} - \cos^{2} (θ) - \sin^{2} (θ)$ .

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Step 9.1

Factor $- 1$ out of $- \cos^{2} (θ)$ .

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

Step 9.2

Factor $- 1$ out of $- \sin^{2} (θ)$ .

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

Step 9.3

Factor $- 1$ out of $- (\cos^{2} (θ)) - (\sin^{2} (θ))$ .

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

Step 9.4

Rearrange terms.

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

Step 9.5

Apply pythagorean identity.

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

Step 9.6

Simplify each term.

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Step 9.6.1

Convert from $\frac{\sin^{2} (θ)}{\cos^{2} (θ)}$ to $\tan^{2} (θ)$ .

$\tan^{2} (θ) - 1 \cdot 1 = 0$

Step 9.6.2

Multiply $- 1$ by $1$ .

$\tan^{2} (θ) - 1 = 0$

Step 10

Solve for $θ$ .

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Step 10.1

Add $1$ to both sides of the equation.

$\tan^{2} (θ) = 1$

Step 10.2

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

$\tan (θ) = \pm \sqrt{1}$

Step 10.3

Any root of $1$ is $1$ .

$\tan (θ) = \pm 1$

Step 10.4

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

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Step 10.4.1

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

$\tan (θ) = 1$

Step 10.4.2

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

$\tan (θ) = - 1$

Step 10.4.3

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

$\tan (θ) = 1, - 1$

Step 10.5

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

$\tan (θ) = 1$

$\tan (θ) = - 1$

Step 10.6

Solve for $θ$ in $\tan (θ) = 1$ .

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Step 10.6.1

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

$θ = \arctan (1)$

Step 10.6.2

Simplify the right side.

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Step 10.6.2.1

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

$θ = \frac{π}{4}$

Step 10.6.3

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.

$θ = π + \frac{π}{4}$

Step 10.6.4

Simplify $π + \frac{π}{4}$ .

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Step 10.6.4.1

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

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

Step 10.6.4.2

Combine fractions.

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Step 10.6.4.2.1

Combine $π$ and $\frac{4}{4}$ .

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

Step 10.6.4.2.2

Combine the numerators over the common denominator.

$θ = \frac{π \cdot 4 + π}{4}$

Step 10.6.4.3

Simplify the numerator.

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Step 10.6.4.3.1

Move $4$ to the left of $π$ .

$θ = \frac{4 \cdot π + π}{4}$

Step 10.6.4.3.2

Add $4 π$ and $π$ .

$θ = \frac{5 π}{4}$

Step 10.6.5

Find the period of $\tan (θ)$ .

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Step 10.6.5.1

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

$\frac{π}{| b |}$

Step 10.6.5.2

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

$\frac{π}{| 1 |}$

Step 10.6.5.3

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

$\frac{π}{1}$

Step 10.6.5.4

Divide $π$ by $1$ .

$π$

Step 10.6.6

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

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

Step 10.7

Solve for $θ$ in $\tan (θ) = - 1$ .

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Step 10.7.1

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

$θ = \arctan (- 1)$

Step 10.7.2

Simplify the right side.

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Step 10.7.2.1

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

$θ = - \frac{π}{4}$

Step 10.7.3

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.

$θ = - \frac{π}{4} - π$

Step 10.7.4

Simplify the expression to find the second solution.

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Step 10.7.4.1

Add $2 π$ to $- \frac{π}{4} - π$ .

$θ = - \frac{π}{4} - π + 2 π$

Step 10.7.4.2

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

$θ = \frac{3 π}{4}$

Step 10.7.5

Find the period of $\tan (θ)$ .

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Step 10.7.5.1

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

$\frac{π}{| b |}$

Step 10.7.5.2

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

$\frac{π}{| 1 |}$

Step 10.7.5.3

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

$\frac{π}{1}$

Step 10.7.5.4

Divide $π$ by $1$ .

$π$

Step 10.7.6

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

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Step 10.7.6.1

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

$- \frac{π}{4} + π$

Step 10.7.6.2

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

$π \cdot \frac{4}{4} - \frac{π}{4}$

Step 10.7.6.3

Combine fractions.

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Step 10.7.6.3.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} - \frac{π}{4}$

Step 10.7.6.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 - π}{4}$

Step 10.7.6.4

Simplify the numerator.

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Step 10.7.6.4.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π - π}{4}$

Step 10.7.6.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{4}$

Step 10.7.6.5

List the new angles.

$θ = \frac{3 π}{4}$

Step 10.7.7

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

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

Step 10.8

List all of the solutions.

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

Step 10.9

Consolidate the answers.

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