Verify the Identity 2cot(x)csc(x)=1/(sec(x)-1)+1/(sec(x)+1)

$2 \cot (x) \csc (x) = \frac{1}{\sec (x) - 1} + \frac{1}{\sec (x) + 1}$

Start on the right side.

$\frac{1}{\sec (x) - 1} + \frac{1}{\sec (x) + 1}$

Add fractions.

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

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

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

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

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

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Multiply $\frac{1}{\sec (x) - 1}$ and $\frac{\sec (x) + 1}{\sec (x) + 1}$ .

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

Multiply $\frac{1}{\sec (x) + 1}$ and $\frac{\sec (x) - 1}{\sec (x) - 1}$ .

$\frac{\sec (x) + 1}{(\sec (x) - 1) (\sec (x) + 1)} + \frac{\sec (x) - 1}{(\sec (x) + 1) (\sec (x) - 1)}$

Reorder the factors of $(\sec (x) - 1) (\sec (x) + 1)$ .

$\frac{\sec (x) + 1}{(\sec (x) + 1) (\sec (x) - 1)} + \frac{\sec (x) - 1}{(\sec (x) + 1) (\sec (x) - 1)}$

Combine the numerators over the common denominator.

$\frac{\sec (x) + 1 + \sec (x) - 1}{(\sec (x) + 1) (\sec (x) - 1)}$

Simplify numerator.

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Add $\sec (x)$ and $\sec (x)$ .

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

Subtract $1$ from $1$ .

$\frac{2 \sec (x) + 0}{(\sec (x) + 1) (\sec (x) - 1)}$

Add $2 \sec (x)$ and $0$ .

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

Simplify denominator.

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Expand $(\sec (x) + 1) (\sec (x) - 1)$ using the FOIL Method.

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

$\frac{2 \sec (x)}{\sec^{2} (x) - 1}$

Apply pythagorean identity.

$\frac{2 \sec (x)}{\tan^{2} (x)}$

Convert to sines and cosines.

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

$\frac{2 \frac{1}{\cos (x)}}{\tan^{2} (x)}$

Write $\tan (x)$ in sines and cosines using the quotient identity.

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

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

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

Simplify.

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Multiply the numerator by the reciprocal of the denominator.

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

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

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

Combine.

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

Cancel the common factor of ${\cos (x)}^{2}$ and $\cos (x)$ .

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

Rewrite $\frac{2 \cos (x)}{\sin^{2} (x)}$ as $2 \cot (x) \csc (x)$ .

$2 \cot (x) \csc (x)$

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

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

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Name

Name	two billion one hundred thirteen million four hundred four thousand fifty-eight

Interesting facts

2113404058 has 16 divisors, whose sum is 3311268480
The reverse of 2113404058 is 8504043112
Previous prime number is 103

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 28
Digital Root 1

Name

Name	one billion nine hundred fifty-five million eight hundred sixty-one thousand six hundred thirty

Interesting facts

1955861630 has 16 divisors, whose sum is 3551708304
The reverse of 1955861630 is 0361685591
Previous prime number is 113

Basic properties

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

Name

Name	three hundred eighty million eighty-five thousand two hundred eighty-nine

Interesting facts

380085289 has 4 divisors, whose sum is 383572420
The reverse of 380085289 is 982580083
Previous prime number is 109

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 43
Digital Root 7

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