Verify the Identity sin(3x)=(sin(x))(4cos(x)^2-1)

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Trigonometry

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

Start on the right side.

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

Apply Pythagorean identity in reverse.

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

Apply the distributive property.

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

Simplify each term.

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

Apply the distributive property.

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

Simplify.

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Simplify each term.

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

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

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

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

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

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

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

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

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

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

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

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

Apply the sine triple-angle identity.

$\sin (3 x)$

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

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Name

Name	three hundred eighty-one million one hundred thirty thousand nine hundred seventy-nine

Interesting facts

381130979 has 4 divisors, whose sum is 381609984
The reverse of 381130979 is 979031183
Previous prime number is 797

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 41
Digital Root 5

Name

Name	two hundred forty-eight million eighty-one thousand five hundred thirty-four

Interesting facts

248081534 has 16 divisors, whose sum is 381701760
The reverse of 248081534 is 435180842
Previous prime number is 439

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 35
Digital Root 8

Name

Name	one billion four hundred twenty-five million six hundred thirty-three thousand eight hundred fifty-two

Interesting facts

1425633852 has 64 divisors, whose sum is 4429987200
The reverse of 1425633852 is 2583365241
Previous prime number is 971

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 39
Digital Root 3

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