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Verify:  `(e^(x)+e^(-x)/2)^(2) - (e^(x)-e^(-x)/2)^(2) = 2`  

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user354582 | Honors

Posted June 20, 2013 at 6:36 AM via web

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Verify:  `(e^(x)+e^(-x)/2)^(2) - (e^(x)-e^(-x)/2)^(2) = 2`

 

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted June 20, 2013 at 6:48 AM (Answer #1)

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The identity `(e^(x)+e^(-x)/2)^(2) - (e^(x)-e^(-x)/2)^(2) = 2` has to be verified.

`(e^(x)+e^(-x)/2)^(2) - (e^(x)-e^(-x)/2)^(2)`

Use the expansion (a + b)^2 = a^2 + b^2 + 2*a*b

= `e^(2x)+e^(-2x)/4 + 1 - e^(2x) - e^(-2x)/4 + 1`

= 2

This proves that `(e^(x)+e^(-x)/2)^(2) - (e^(x)-e^(-x)/2)^(2) = 2`

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aruv | High School Teacher | Valedictorian

Posted June 20, 2013 at 8:28 AM (Answer #2)

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Use an identity `A^2-B^2=(A-B)(A+B)`

`LHS=(e^x+e^(-x)/2)^2-(e^x-e^(-x)/2)^2`

`=(e^x+e^(-x)/2-e^x+e^(-x)/2)(e^x+e^(-x)/2+e^x-e^(-x)/2)`

`=(e^(-x))(2e^x)`

`=2e^(-x+x)`

`=2e^0`

`=2`

`=RHS`

So qed.

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