Precalculus Examples

$\tan (X) \cos (X) = \cos (X)$

Step 1

Simplify the left side.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

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

$\frac{\sin (X)}{\cos (X)} \cos (X) = \cos (X)$

Step 1.1.2

Cancel the common factors.

$\sin (X) = \cos (X)$

Step 2

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

$\frac{\sin (X)}{\cos (X)} = \frac{\cos (X)}{\cos (X)}$

Step 3

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

$\tan (X) = \frac{\cos (X)}{\cos (X)}$

Step 4

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

Tap for more steps...

Step 4.1

Cancel the common factor.

$\tan (X) = \frac{\cos (X)}{\cos (X)}$

Step 4.2

Rewrite the expression.

$\tan (X) = 1$

Step 5

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

$X = \arctan (1)$

Step 6

Simplify the right side.

Tap for more steps...

Step 6.1

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

$X = \frac{π}{4}$

Step 7

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 8

Simplify $π + \frac{π}{4}$ .

Tap for more steps...

Step 8.1

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 8.2

Combine fractions.

Tap for more steps...

Step 8.2.1

Combine $π$ and $\frac{4}{4}$ .

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

Step 8.2.2

Combine the numerators over the common denominator.

$X = \frac{π \cdot 4 + π}{4}$

Step 8.3

Simplify the numerator.

Tap for more steps...

Step 8.3.1

Move $4$ to the left of $π$ .

$X = \frac{4 \cdot π + π}{4}$

Step 8.3.2

Add $4 π$ and $π$ .

$X = \frac{5 π}{4}$

Step 9

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

Tap for more steps...

Step 9.1

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

$\frac{π}{| b |}$

Step 9.2

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

$\frac{π}{| 1 |}$

Step 9.3

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

$\frac{π}{1}$

Step 9.4

Divide $π$ by $1$ .

$π$

Step 10

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 11

Consolidate the answers.

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