Precalculus Examples

$\tan (2 x) = \tan (x)$

Step 1

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

$\tan (2 x) - \tan (x) = 0$

Step 2

Simplify each term.

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Step 2.1

Apply the tangent double-angle identity.

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

Step 2.2

Simplify the denominator.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 3

Factor $\tan (x)$ out of $\frac{2 \tan (x)}{(1 + \tan (x)) (1 - \tan (x))} - \tan (x)$ .

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Step 3.1

Factor $\tan (x)$ out of $\frac{2 \tan (x)}{(1 + \tan (x)) (1 - \tan (x))}$ .

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

Step 3.2

Factor $\tan (x)$ out of $- \tan (x)$ .

$\tan (x) \frac{2}{(1 + \tan (x)) (1 - \tan (x))} + \tan (x) \cdot - 1 = 0$

Step 3.3

Factor $\tan (x)$ out of $\tan (x) \frac{2}{(1 + \tan (x)) (1 - \tan (x))} + \tan (x) \cdot - 1$ .

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

Step 4

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

$\tan (x) = 0$

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

Step 5

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

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Step 5.1

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

$\tan (x) = 0$

Step 5.2

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

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Step 5.2.1

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

$x = \arctan (0)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

The exact value of $\arctan (0)$ is $0$ .

$x = 0$

Step 5.2.3

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.

$x = π + 0$

Step 5.2.4

Add $π$ and $0$ .

$x = π$

Step 5.2.5

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

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Step 5.2.5.1

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

$\frac{π}{| b |}$

Step 5.2.5.2

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

$\frac{π}{| 1 |}$

Step 5.2.5.3

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

$\frac{π}{1}$

Step 5.2.5.4

Divide $π$ by $1$ .

$π$

Step 5.2.6

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

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

Step 6

Set $\frac{2}{(1 + \tan (x)) (1 - \tan (x))} - 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $\frac{2}{(1 + \tan (x)) (1 - \tan (x))} - 1$ equal to $0$ .

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

Step 6.2

Solve $\frac{2}{(1 + \tan (x)) (1 - \tan (x))} - 1 = 0$ for $x$ .

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Step 6.2.1

Find the LCD of the terms in the equation.

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Step 6.2.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$(1 + \tan (x)) (1 - \tan (x)), 1, 1$

Step 6.2.1.2

The LCM of one and any expression is the expression.

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

Step 6.2.2

Multiply each term in $\frac{2}{(1 + \tan (x)) (1 - \tan (x))} - 1 = 0$ by $(1 + \tan (x)) (1 - \tan (x))$ to eliminate the fractions.

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Step 6.2.2.1

Multiply each term in $\frac{2}{(1 + \tan (x)) (1 - \tan (x))} - 1 = 0$ by $(1 + \tan (x)) (1 - \tan (x))$ .

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

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Simplify each term.

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Step 6.2.2.2.1.1

Cancel the common factor of $(1 + \tan (x)) (1 - \tan (x))$ .

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Step 6.2.2.2.1.1.1

Cancel the common factor.

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

Step 6.2.2.2.1.1.2

Rewrite the expression.

$2 - ((1 + \tan (x)) (1 - \tan (x))) = 0 ((1 + \tan (x)) (1 - \tan (x)))$

Step 6.2.2.2.1.2

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

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Step 6.2.2.2.1.2.1

Apply the distributive property.

$2 - (1 (1 - \tan (x)) + \tan (x) (1 - \tan (x))) = 0 ((1 + \tan (x)) (1 - \tan (x)))$

Step 6.2.2.2.1.2.2

Apply the distributive property.

$2 - (1 \cdot 1 + 1 (- \tan (x)) + \tan (x) (1 - \tan (x))) = 0 ((1 + \tan (x)) (1 - \tan (x)))$

Step 6.2.2.2.1.2.3

Apply the distributive property.

$2 - (1 \cdot 1 + 1 (- \tan (x)) + \tan (x) \cdot 1 + \tan (x) (- \tan (x))) = 0 ((1 + \tan (x)) (1 - \tan (x)))$

Step 6.2.2.2.1.3

Simplify and combine like terms.

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Step 6.2.2.2.1.3.1

Simplify each term.

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Step 6.2.2.2.1.3.1.1

Multiply $1$ by $1$ .

$2 - (1 + 1 (- \tan (x)) + \tan (x) \cdot 1 + \tan (x) (- \tan (x))) = 0 ((1 + \tan (x)) (1 - \tan (x)))$

Step 6.2.2.2.1.3.1.2

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

$2 - (1 - \tan (x) + \tan (x) \cdot 1 + \tan (x) (- \tan (x))) = 0 ((1 + \tan (x)) (1 - \tan (x)))$

Step 6.2.2.2.1.3.1.3

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

$2 - (1 - \tan (x) + \tan (x) + \tan (x) (- \tan (x))) = 0 ((1 + \tan (x)) (1 - \tan (x)))$

Step 6.2.2.2.1.3.1.4

Rewrite using the commutative property of multiplication.

$2 - (1 - \tan (x) + \tan (x) - \tan (x) \tan (x)) = 0 ((1 + \tan (x)) (1 - \tan (x)))$

Step 6.2.2.2.1.3.1.5

Multiply $\tan (x)$ by $\tan (x)$ by adding the exponents.

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Step 6.2.2.2.1.3.1.5.1

Move $\tan (x)$ .

$2 - (1 - \tan (x) + \tan (x) - (\tan (x) \tan (x))) = 0 ((1 + \tan (x)) (1 - \tan (x)))$

Step 6.2.2.2.1.3.1.5.2

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

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

Step 6.2.2.2.1.3.2

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

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

Step 6.2.2.2.1.3.3

Add $1$ and $0$ .

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

Step 6.2.2.2.1.4

Apply the distributive property.

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

Step 6.2.2.2.1.5

Multiply $- 1$ by $1$ .

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

Step 6.2.2.2.1.6

Multiply $- - \tan^{2} (x)$ .

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Step 6.2.2.2.1.6.1

Multiply $- 1$ by $- 1$ .

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

Step 6.2.2.2.1.6.2

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

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

Step 6.2.2.2.2

Subtract $1$ from $2$ .

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

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

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

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Step 6.2.2.3.1.1

Apply the distributive property.

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

Step 6.2.2.3.1.2

Apply the distributive property.

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

Step 6.2.2.3.1.3

Apply the distributive property.

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

Step 6.2.2.3.2

Simplify and combine like terms.

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Step 6.2.2.3.2.1

Simplify each term.

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Step 6.2.2.3.2.1.1

Multiply $1$ by $1$ .

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

Step 6.2.2.3.2.1.2

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

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

Step 6.2.2.3.2.1.3

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

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

Step 6.2.2.3.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 6.2.2.3.2.1.5

Multiply $\tan (x)$ by $\tan (x)$ by adding the exponents.

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Step 6.2.2.3.2.1.5.1

Move $\tan (x)$ .

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

Step 6.2.2.3.2.1.5.2

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

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

Step 6.2.2.3.2.2

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

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

Step 6.2.2.3.2.3

Add $1$ and $0$ .

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

Step 6.2.2.3.3

Multiply $0$ by $1 - \tan^{2} (x)$ .

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

Step 6.2.3

Solve the equation.

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Step 6.2.3.1

Subtract $1$ from both sides of the equation.

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

Step 6.2.3.2

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

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

Step 6.2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$\tan (x) = \pm i$

Step 6.2.3.4

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

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Step 6.2.3.4.1

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

$\tan (x) = i$

Step 6.2.3.4.2

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

$\tan (x) = - i$

Step 6.2.3.4.3

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

$\tan (x) = i, - i$

Step 6.2.4

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

$\tan (x) = i$

$\tan (x) = - i$

Step 6.2.5

Solve for $x$ in $\tan (x) = i$ .

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Step 6.2.5.1

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

$x = \arctan (i)$

Step 6.2.5.2

The inverse tangent of $\arctan (i)$ is undefined.

Undefined

Step 6.2.6

Solve for $x$ in $\tan (x) = - i$ .

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Step 6.2.6.1

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

$x = \arctan (- i)$

Step 6.2.6.2

The inverse tangent of $\arctan (- i)$ is undefined.

Undefined

Step 6.2.7

List all of the solutions.

No solution

Step 7

The final solution is all the values that make $\tan (x) (\frac{2}{(1 + \tan (x)) (1 - \tan (x))} - 1) = 0$ true.

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

Step 8

Consolidate the answers.

$x = π n$ , for any integer $n$