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	one billion eight hundred seven million eight hundred thirty-seven thousand nine hundred twenty-nine

Interesting facts

1807837929 has 8 divisors, whose sum is 2009392000
The reverse of 1807837929 is 9297387081
Previous prime number is 3079

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 54
Digital Root 9

Name

Name	three hundred seventy-eight million eight hundred eighty-six thousand nine hundred eighty

Interesting facts

378886980 has 64 divisors, whose sum is 963532800
The reverse of 378886980 is 089688873
Previous prime number is 19

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 57
Digital Root 3

Name

Name	one billion six hundred ninety-eight million five hundred eighty-six thousand eight hundred ninety-five

Interesting facts

1698586895 has 4 divisors, whose sum is 2038304280
The reverse of 1698586895 is 5986858961
Previous prime number is 5

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 65
Digital Root 2

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