Trigonometry Examples

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

Step 1

Replace the $\sin^{2} (x)$ with $1 - \cos^{2} (x)$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

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

Step 2

Apply the distributive property.

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

Step 3

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

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

Step 4

Multiply $\cos^{2} (x)$ by $\cos (x)$ by adding the exponents.

Tap for more steps...

Step 4.1

Move $\cos (x)$ .

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

Step 4.2

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

Tap for more steps...

Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

Add $1$ and $2$ .

$\cos (x) - \cos^{3} (x) = \cos (x)$

Step 5

Reorder the polynomial.

$- \cos^{3} (x) + \cos (x) = \cos (x)$

Step 6

Move all terms containing $\cos (x)$ to the left side of the equation.

Tap for more steps...

Step 6.1

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

$- \cos^{3} (x) + \cos (x) - \cos (x) = 0$

Step 6.2

Combine the opposite terms in $- \cos^{3} (x) + \cos (x) - \cos (x)$ .

Tap for more steps...

Step 6.2.1

Subtract $\cos (x)$ from $\cos (x)$ .

$- \cos^{3} (x) + 0 = 0$

Step 6.2.2

Add $- \cos^{3} (x)$ and $0$ .

$- \cos^{3} (x) = 0$

Step 7

Divide each term in $- \cos^{3} (x) = 0$ by $- 1$ and simplify.

Tap for more steps...

Step 7.1

Divide each term in $- \cos^{3} (x) = 0$ by $- 1$ .

$\frac{- \cos^{3} (x)}{- 1} = \frac{0}{- 1}$

Step 7.2

Simplify the left side.

Tap for more steps...

Step 7.2.1

Dividing two negative values results in a positive value.

$\frac{\cos^{3} (x)}{1} = \frac{0}{- 1}$

Step 7.2.2

Divide $\cos^{3} (x)$ by $1$ .

$\cos^{3} (x) = \frac{0}{- 1}$

Step 7.3

Simplify the right side.

Tap for more steps...

Step 7.3.1

Divide $0$ by $- 1$ .

$\cos^{3} (x) = 0$

Step 8

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

$\cos (x) = \sqrt[3]{0}$

Step 9

Simplify $\sqrt[3]{0}$ .

Tap for more steps...

Step 9.1

Rewrite $0$ as $0^{3}$ .

$\cos (x) = \sqrt[3]{0^{3}}$

Step 9.2

Pull terms out from under the radical, assuming real numbers.

$\cos (x) = 0$

Step 10

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

$x = \arccos (0)$

Step 11

Simplify the right side.

Tap for more steps...

Step 11.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 12

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{2}$

Step 13

Simplify $2 π - \frac{π}{2}$ .

Tap for more steps...

Step 13.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 13.2

Combine fractions.

Tap for more steps...

Step 13.2.1

Combine $2 π$ and $\frac{2}{2}$ .

$x = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 13.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 2 - π}{2}$

Step 13.3

Simplify the numerator.

Tap for more steps...

Step 13.3.1

Multiply $2$ by $2$ .

$x = \frac{4 π - π}{2}$

Step 13.3.2

Subtract $π$ from $4 π$ .

$x = \frac{3 π}{2}$

Step 14

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

Tap for more steps...

Step 14.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 14.2

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

$\frac{2 π}{| 1 |}$

Step 14.3

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

$\frac{2 π}{1}$

Step 14.4

Divide $2 π$ by $1$ .

$2 π$

Step 15

The period of the $\cos (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 16

Consolidate the answers.

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