Trigonometry Examples

$\sin (4 x) \sin (2 x)$

Step 1

Factor $2$ out of $4 x$ .

$\sin (2 (2 x)) \sin (2 x)$

Step 2

Apply the sine double-angle identity.

$2 (2 \sin (x) \cos (x)) \cos (2 x) \sin (2 x)$

Step 3

Multiply $2$ by $2$ .

$4 (\sin (x) \cos (x)) \cos (2 x) \sin (2 x)$

Step 4

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

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

Step 5

Apply the distributive property.

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

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.3

Add $2$ and $1$ .

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

Step 7

Multiply $- 1$ by $4$ .

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

Step 8

Multiply $2$ by $4$ .

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) \sin (2 x)$

Step 9

Apply the sine double-angle identity.

$(8 \sin (x) \cos^{3} (x) - 4 \sin (x) \cos (x)) (2 \sin (x) \cos (x))$

Step 10

Apply the distributive property.

$8 \sin (x) \cos^{3} (x) (2 \sin (x) \cos (x)) - 4 \sin (x) \cos (x) (2 \sin (x) \cos (x))$

Step 11

Multiply $\cos^{3} (x)$ by $\cos (x)$ by adding the exponents.

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Step 11.1

Move $\cos (x)$ .

$8 \sin (x) (\cos (x) \cos^{3} (x)) (2 \sin (x)) - 4 \sin (x) \cos (x) (2 \sin (x) \cos (x))$

Step 11.2

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

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Step 11.2.1

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

$8 \sin (x) (\cos^{1} (x) \cos^{3} (x)) (2 \sin (x)) - 4 \sin (x) \cos (x) (2 \sin (x) \cos (x))$

Step 11.2.2

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

$8 \sin (x) {\cos (x)}^{1 + 3} (2 \sin (x)) - 4 \sin (x) \cos (x) (2 \sin (x) \cos (x))$

Step 11.3

Add $1$ and $3$ .

$8 \sin (x) \cos^{4} (x) (2 \sin (x)) - 4 \sin (x) \cos (x) (2 \sin (x) \cos (x))$

Step 12

Multiply $- 4 \sin (x) \cos (x) (2 \sin (x) \cos (x))$ .

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Step 12.1

Multiply $2$ by $- 4$ .

$8 \sin (x) \cos^{4} (x) (2 \sin (x)) - 8 \sin (x) \cos (x) (\sin (x) \cos (x))$

Step 12.2

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

$8 \sin (x) \cos^{4} (x) (2 \sin (x)) - 8 (\sin^{1} (x) \sin (x)) \cos (x) \cos (x)$

Step 12.3

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

$8 \sin (x) \cos^{4} (x) (2 \sin (x)) - 8 (\sin^{1} (x) \sin^{1} (x)) \cos (x) \cos (x)$

Step 12.4

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

$8 \sin (x) \cos^{4} (x) (2 \sin (x)) - 8 {\sin (x)}^{1 + 1} \cos (x) \cos (x)$

Step 12.5

Add $1$ and $1$ .

$8 \sin (x) \cos^{4} (x) (2 \sin (x)) - 8 \sin^{2} (x) \cos (x) \cos (x)$

Step 12.6

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

$8 \sin (x) \cos^{4} (x) (2 \sin (x)) - 8 \sin^{2} (x) (\cos^{1} (x) \cos (x))$

Step 12.7

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

$8 \sin (x) \cos^{4} (x) (2 \sin (x)) - 8 \sin^{2} (x) (\cos^{1} (x) \cos^{1} (x))$

Step 12.8

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

$8 \sin (x) \cos^{4} (x) (2 \sin (x)) - 8 \sin^{2} (x) {\cos (x)}^{1 + 1}$

Step 12.9

Add $1$ and $1$ .

$8 \sin (x) \cos^{4} (x) (2 \sin (x)) - 8 \sin^{2} (x) \cos^{2} (x)$

Step 13

Multiply $8 \sin (x) \cos^{4} (x) (2 \sin (x))$ .

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Step 13.1

Multiply $2$ by $8$ .

$16 \sin (x) \cos^{4} (x) \sin (x) - 8 \sin^{2} (x) \cos^{2} (x)$

Step 13.2

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

$16 (\sin^{1} (x) \sin (x)) \cos^{4} (x) - 8 \sin^{2} (x) \cos^{2} (x)$

Step 13.3

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

$16 (\sin^{1} (x) \sin^{1} (x)) \cos^{4} (x) - 8 \sin^{2} (x) \cos^{2} (x)$

Step 13.4

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

$16 {\sin (x)}^{1 + 1} \cos^{4} (x) - 8 \sin^{2} (x) \cos^{2} (x)$

Step 13.5

Add $1$ and $1$ .

$16 \sin^{2} (x) \cos^{4} (x) - 8 \sin^{2} (x) \cos^{2} (x)$