Precalculus Examples

$\frac{\sin (4 x)}{\sin (x)}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $2$ out of $4 x$ .

$\frac{\sin (2 (2 x))}{\sin (x)}$

Step 1.2

Apply the sine double-angle identity.

$\frac{2 (2 \sin (x) \cos (x)) \cos (2 x)}{\sin (x)}$

Step 1.3

Use the double-angle identity to transform $\cos (2 x)$ to $2 \cos^{2} (x) - 1$ .

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

Step 1.4

Multiply $2$ by $2$ .

$\frac{4 \sin (x) \cos (x) (2 \cos^{2} (x) - 1)}{\sin (x)}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $\sin (x)$ .

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Step 2.1.1

Cancel the common factor.

$\frac{4 \sin (x) \cos (x) (2 \cos^{2} (x) - 1)}{\sin (x)}$

Step 2.1.2

Divide $4 \cos (x) (2 \cos^{2} (x) - 1)$ by $1$ .

$4 \cos (x) (2 \cos^{2} (x) - 1)$

Step 2.2

Apply the distributive property.

$4 \cos (x) (2 \cos^{2} (x)) + 4 \cos (x) \cdot - 1$

Step 3

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

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Step 3.1

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

$4 (\cos^{2} (x) \cos (x)) \cdot 2 + 4 \cos (x) \cdot - 1$

Step 3.2

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

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Step 3.2.1

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

$4 (\cos^{2} (x) \cos^{1} (x)) \cdot 2 + 4 \cos (x) \cdot - 1$

Step 3.2.2

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

$4 {\cos (x)}^{2 + 1} \cdot 2 + 4 \cos (x) \cdot - 1$

Step 3.3

Add $2$ and $1$ .

$4 \cos^{3} (x) \cdot 2 + 4 \cos (x) \cdot - 1$

Step 4

Multiply $- 1$ by $4$ .

$4 \cos^{3} (x) \cdot 2 - 4 \cos (x)$

Step 5

Multiply $2$ by $4$ .

$8 \cos^{3} (x) - 4 \cos (x)$