Verify the Identity (1-cos(x))/(sin(x))+(sin(x))/(1-cos(x))=2csc(x)

$\frac{1 - \cos (x)}{\sin (x)} + \frac{\sin (x)}{1 - \cos (x)} = 2 \csc (x)$

Start on the left side.

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

Add fractions.

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To write $\frac{1 - \cos (x)}{\sin (x)}$ as a fraction with a common denominator, multiply by $\frac{1 - \cos (x)}{1 - \cos (x)}$ .

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

To write $\frac{\sin (x)}{1 - \cos (x)}$ as a fraction with a common denominator, multiply by $\frac{\sin (x)}{\sin (x)}$ .

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

Write each expression with a common denominator of $\sin (x) (1 - \cos (x))$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

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

Combine.

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

Reorder the factors of $(1 - \cos (x)) \sin (x)$ .

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

Combine the numerators over the common denominator.

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

Simplify each term.

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

Simplify denominator.

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Apply the distributive property.

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

Multiply $\sin (x)$ by $1$ .

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

Simplify each term.

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

Apply pythagorean identity.

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

Simplify.

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Simplify the numerator.

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Add $1$ and $1$ .

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

Factor $2$ out of $2 - 2 \cos (x)$ .

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Factor $2$ out of $2$ .

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

Factor $2$ out of $- 2 \cos (x)$ .

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

Factor $2$ out of $2 (1) + 2 (- \cos (x))$ .

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

Factor $\sin (x)$ out of $\sin (x) - \sin (x) \cos (x)$ .

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Multiply by $1$ .

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

Factor $\sin (x)$ out of $- \sin (x) \cos (x)$ .

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

Factor $\sin (x)$ out of $\sin (x) \cdot 1 + \sin (x) (- \cos (x))$ .

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

Cancel the common factor of $1 - \cos (x)$ .

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

Now consider the right side of the equation.

$2 \csc (x)$

Apply the reciprocal identity to $\csc (x)$ .

$2 \frac{1}{\sin (x)}$

Combine $2$ and $\frac{1}{\sin (x)}$ .

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

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

$\frac{1 - \cos (x)}{\sin (x)} + \frac{\sin (x)}{1 - \cos (x)} = 2 \csc (x)$ is an identity

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Name

Name	one billion four hundred sixty-seven million four hundred forty-two thousand five hundred seventy-three

Interesting facts

1467442573 has 16 divisors, whose sum is 1554973056
The reverse of 1467442573 is 3752447641
Previous prime number is 83

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 43
Digital Root 7

Name

Name	one billion three hundred fifty-two million three hundred fifty-eight thousand eight hundred seventy-two

Interesting facts

1352358872 has 32 divisors, whose sum is 4626736632
The reverse of 1352358872 is 2788532531
Previous prime number is 73

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 44
Digital Root 8

Name

Name	nine hundred forty-eight million four hundred forty-three thousand six hundred ninety-four

Interesting facts

948443694 has 8 divisors, whose sum is 1465776720
The reverse of 948443694 is 496344849
Previous prime number is 33

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 51
Digital Root 6

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