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

Verify the Identity 1/(1+cos(x))-1/(1-cos(x))=-2csc(x)cot(x)
Start on the left side.
Subtract fractions.
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To write as a fraction with a common denominator, multiply by .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Combine.
Combine.
Reorder the factors of .
Combine the numerators over the common denominator.
Simplify numerator.
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Simplify each term.
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Multiply by .
Apply the distributive property.
Multiply by .
Subtract from .
Subtract from .
Subtract from .
Simplify denominator.
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Expand using the FOIL Method.
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Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Apply pythagorean identity.
Move the negative in front of the fraction.
Now consider the right side of the equation.
Convert to sines and cosines.
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Apply the reciprocal identity to .
Write in sines and cosines using the quotient identity.
Simplify.
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Combine and .
Move the negative in front of the fraction.
Multiply .
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Multiply and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Move to the left of .
Because the two sides have been shown to be equivalent, the equation is an identity.
is an identity
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Name

Name three hundred ninety-one million twenty-three thousand six hundred fifty-seven

Interesting facts

  • 391023657 has 16 divisors, whose sum is 440880000
  • The reverse of 391023657 is 756320193
  • Previous prime number is 109

Basic properties

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

Name

Name two hundred three million sixty-nine thousand eight hundred twenty-four

Interesting facts

  • 203069824 has 512 divisors, whose sum is 3480846696
  • The reverse of 203069824 is 428960302
  • Previous prime number is 331

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 9
  • Sum of Digits 34
  • Digital Root 7

Name

Name one billion seven hundred seventy-eight million sixteen thousand four hundred five

Interesting facts

  • 1778016405 has 32 divisors, whose sum is 3105641088
  • The reverse of 1778016405 is 5046108771
  • Previous prime number is 5741

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 10
  • Sum of Digits 39
  • Digital Root 3