Trigonometry Examples

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

Step 1

Start on the right side.

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

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

Apply the distributive property.

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

Step 4

Simplify each term.

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

Step 5

Apply the distributive property.

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

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

Move $4$ to the left of $\sin (x)$ .

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

Step 6.1.2

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

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Step 6.1.2.1

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

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

Step 6.1.2.2

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

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Step 6.1.2.2.1

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

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

Step 6.1.2.2.2

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

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

Step 6.1.2.3

Add $2$ and $1$ .

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

Step 6.1.3

Move $- 4$ to the left of ${\sin (x)}^{3}$ .

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

Step 6.1.4

Move $- 1$ to the left of $\sin (x)$ .

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

Step 6.1.5

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

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

Step 6.2

Subtract $\sin (x)$ from $4 \sin (x)$ .

$- 4 \sin^{3} (x) + 3 \sin (x)$

Step 7

Apply the sine triple-angle identity.

$\sin (3 x)$

Step 8

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

$\sin (3 x) = (\sin (x)) (4 \cos^{2} (x) - 1)$ is an identity