Algebra Examples

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

Step 1

Simplify the left side.

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Step 1.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 1.1.1

Reorder $\cos (x)$ and $\tan (x)$ .

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

Step 1.1.2

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

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

Step 1.1.3

Cancel the common factors.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Divide $- 1$ by $1$ .

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

Step 5

Separate fractions.

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

Step 6

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

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

Step 7

Divide $0$ by $1$ .

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

Step 8

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

$\tan (x) - 1 = 0$

Step 9

Add $1$ to both sides of the equation.

$\tan (x) = 1$

Step 10

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

$x = \arctan (1)$

Step 11

Simplify the right side.

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Step 11.1

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

$x = \frac{π}{4}$

Step 12

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 13

Simplify $π + \frac{π}{4}$ .

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Step 13.1

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 13.2

Combine fractions.

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Step 13.2.1

Combine $π$ and $\frac{4}{4}$ .

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

Step 13.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 4 + π}{4}$

Step 13.3

Simplify the numerator.

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Step 13.3.1

Move $4$ to the left of $π$ .

$x = \frac{4 \cdot π + π}{4}$

Step 13.3.2

Add $4 π$ and $π$ .

$x = \frac{5 π}{4}$

Step 14

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

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Step 14.1

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

$\frac{π}{| b |}$

Step 14.2

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

$\frac{π}{| 1 |}$

Step 14.3

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

$\frac{π}{1}$

Step 14.4

Divide $π$ by $1$ .

$π$

Step 15

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 16

Consolidate the answers.

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