Precalculus Examples

$\cos^{3} (x) = \cos (x) - \sin^{2} (x) \cos (x)$

Step 1

Start on the right side.

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

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Simplify each term.

Tap for more steps...

Step 3.1.1

Apply the distributive property.

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

Step 3.1.2

Multiply $- 1$ by $1$ .

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

Step 3.1.3

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

Tap for more steps...

Step 3.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.1.3.2

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

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

Step 3.1.4

Apply the distributive property.

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

Step 3.1.5

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

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

Step 3.1.6

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

Tap for more steps...

Step 3.1.6.1

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

Tap for more steps...

Step 3.1.6.1.1

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

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

Step 3.1.6.1.2

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

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

Step 3.1.6.2

Add $2$ and $1$ .

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

Step 3.2

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

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

Step 3.3

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

$\cos^{3} (x)$

Step 4

Because the two sides have been shown to be equivalent, the equation is an identity.

$\cos^{3} (x) = \cos (x) - \sin^{2} (x) \cos (x)$ is an identity