Verify the Identity ((1+cos(A))(1-cos(A)))/(sin(A))=sin(A)

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Trigonometry

$\frac{(1 + \cos (A)) (1 - \cos (A))}{\sin (A)} = \sin (A)$

Start on the left side.

$\frac{(1 + \cos (A)) (1 - \cos (A))}{\sin (A)}$

Apply the distributive property.

$\frac{(1 + \cos (A)) \cdot 1 + (1 + \cos (A)) (- \cos (A))}{\sin (A)}$

Simplify.

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Simplify each term.

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Multiply $1 + \cos (A)$ by $1$ .

$\frac{1 + \cos (A) + (1 + \cos (A)) (- \cos (A))}{\sin (A)}$

Apply the distributive property.

$\frac{1 + \cos (A) + 1 (- \cos (A)) + \cos (A) (- \cos (A))}{\sin (A)}$

Multiply $- \cos (A)$ by $1$ .

$\frac{1 + \cos (A) - \cos (A) + \cos (A) (- \cos (A))}{\sin (A)}$

Multiply $\cos (A) (- \cos (A))$ .

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

$\frac{1 + \cos (A) - \cos (A) - ({\cos (A)}^{1} \cos (A))}{\sin (A)}$

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

$\frac{1 + \cos (A) - \cos (A) - ({\cos (A)}^{1} {\cos (A)}^{1})}{\sin (A)}$

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

$\frac{1 + \cos (A) - \cos (A) - {\cos (A)}^{1 + 1}}{\sin (A)}$

Add $1$ and $1$ .

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

Subtract $\cos (A)$ from $\cos (A)$ .

$\frac{1 + 0 - {\cos (A)}^{2}}{\sin (A)}$

Add $1$ and $0$ .

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

Apply pythagorean identity.

$\frac{\sin^{2} (A)}{\sin (A)}$

Cancel the common factor of ${\sin (A)}^{2}$ and $\sin (A)$ .

$\sin (A)$

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

$\frac{(1 + \cos (A)) (1 - \cos (A))}{\sin (A)} = \sin (A)$ is an identity

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Name

Name	one billion seventy-four million eight hundred ten thousand eight hundred twenty-two

Interesting facts

1074810822 has 32 divisors, whose sum is 2278098432
The reverse of 1074810822 is 2280184701
Previous prime number is 1303

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 33
Digital Root 6

Name

Name	one billion three hundred ninety-one million three hundred fifty-five thousand three hundred thirty-three

Interesting facts

1391355333 has 8 divisors, whose sum is 2061267200
The reverse of 1391355333 is 3335531931
Previous prime number is 9

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 36
Digital Root 9

Name

Name	one billion seventy-four million nine hundred seventy-seven thousand nine hundred eighteen

Interesting facts

1074977918 has 8 divisors, whose sum is 1842819312
The reverse of 1074977918 is 8197794701
Previous prime number is 7

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 53
Digital Root 8

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Graph 174

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Convert to a Simplified Fraction square root of 64