Verify the Identity (cos(x))/(sec(x)-1)-(cos(x))/(sec(x)+1)=(2cos(x))/(tan(x)^2)

$\frac{\cos (x)}{\sec (x) - 1} - \frac{\cos (x)}{\sec (x) + 1} = \frac{2 \cos (x)}{\tan^{2} (x)}$

Start on the left side.

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

Subtract fractions.

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

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

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

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

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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Combine.

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

Combine.

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

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

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

Combine the numerators over the common denominator.

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

Simplify numerator.

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Simplify each term.

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Apply the distributive property.

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

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

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

Apply the distributive property.

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

Multiply $- \cos (x) \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

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

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

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

Subtract $\cos (x) \sec (x)$ from $\cos (x) \sec (x)$ .

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

Add $0$ and $\cos (x)$ .

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

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

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

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.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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

Apply pythagorean identity.

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

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

$\frac{\cos (x)}{\sec (x) - 1} - \frac{\cos (x)}{\sec (x) + 1} = \frac{2 \cos (x)}{\tan^{2} (x)}$ is an identity

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Name

Name	six hundred thirteen million three hundred forty-one thousand nine hundred thirty-one

Interesting facts

613341931 has 4 divisors, whose sum is 614871864
The reverse of 613341931 is 139143316
Previous prime number is 401

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 31
Digital Root 4

Name

Name	one billion six hundred forty-nine million four hundred eighty thousand four hundred sixteen

Interesting facts

1649480416 has 256 divisors, whose sum is 13034240064
The reverse of 1649480416 is 6140849461
Previous prime number is 47

Basic properties

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

Name

Name	three hundred forty-five million five hundred fifty-one thousand eight hundred sixty-six

Interesting facts

345551866 has 16 divisors, whose sum is 565830720
The reverse of 345551866 is 668155543
Previous prime number is 1777

Basic properties

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

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