Trigonometry Examples

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

Step 1

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = \cos (x)$ and $b = \sin (x)$ .

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

Step 2

Move $\sin^{2} (x)$ .

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

Step 3

Rearrange terms.

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

Step 4

Apply pythagorean identity.

$(\cos (x) - \sin (x)) (1 + \cos (x) \sin (x))$

Step 5

Expand $(\cos (x) - \sin (x)) (1 + \cos (x) \sin (x))$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$\cos (x) (1 + \cos (x) \sin (x)) - \sin (x) (1 + \cos (x) \sin (x))$

Step 5.2

Apply the distributive property.

$\cos (x) \cdot 1 + \cos (x) (\cos (x) \sin (x)) - \sin (x) (1 + \cos (x) \sin (x))$

Step 5.3

Apply the distributive property.

$\cos (x) \cdot 1 + \cos (x) (\cos (x) \sin (x)) - \sin (x) \cdot 1 - \sin (x) (\cos (x) \sin (x))$

Step 6

Simplify each term.

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Step 6.1

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

$\cos (x) + \cos (x) (\cos (x) \sin (x)) - \sin (x) \cdot 1 - \sin (x) (\cos (x) \sin (x))$

Step 6.2

Multiply $\cos (x) (\cos (x) \sin (x))$ .

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

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

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

Step 6.2.4

Add $1$ and $1$ .

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

Step 6.3

Multiply $- 1$ by $1$ .

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

Step 6.4

Multiply $- \sin (x) (\cos (x) \sin (x))$ .

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Step 6.4.1

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

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

Step 6.4.2

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

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

Step 6.4.3

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

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

Step 6.4.4

Add $1$ and $1$ .

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