Precalculus Examples

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

Step 1

Start on the right side.

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

Step 2

Apply Pythagorean identity in reverse.

$\cos^{3} (x) - 3 (1 - \cos^{2} (x)) \cos (x)$

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

Apply the distributive property.

${\cos (x)}^{3} + (- 3 \cdot 1 - 3 (- {\cos (x)}^{2})) \cos (x)$

Step 3.1.2

Multiply $- 3$ by $1$ .

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

Step 3.1.3

Multiply $- 1$ by $- 3$ .

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

Step 3.1.4

Apply the distributive property.

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

Step 3.1.5

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

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Step 3.1.5.1

Move $\cos (x)$ .

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

Step 3.1.5.2

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

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Step 3.1.5.2.1

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

${\cos (x)}^{3} - 3 \cos (x) + 3 ({\cos (x)}^{1} {\cos (x)}^{2})$

Step 3.1.5.2.2

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

${\cos (x)}^{3} - 3 \cos (x) + 3 {\cos (x)}^{1 + 2}$

Step 3.1.5.3

Add $1$ and $2$ .

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

Step 3.2

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

$4 \cos^{3} (x) - 3 \cos (x)$

Step 4

Apply the cosine triple-angle identity.

$\cos (3 x)$

Step 5

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

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