My Math Forum Can anyone tell what algebraic formula/trick is used here?

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 July 28th, 2013, 08:49 AM #1 Senior Member   Joined: Aug 2011 Posts: 149 Thanks: 0 Can anyone tell what algebraic formula/trick is used here? Could anyone tell me step-by-step how to get from left to right? $\dfrac{1}{\sqrt{1-$$\frac{1-x^2}{1+x^2}$$^2}}= \dfrac{|1+x2|}{4x^2}$ I tried to use formula (a-b)(a-b) but didn't get far with it. Could anyone tell me where $\sqrt(4x^2)$ and |1+x^2| come from? Rest I can understand.
 July 28th, 2013, 09:08 AM #2 Senior Member   Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116 Re: Can anyone tell what algebraic formula/trick is used her The trick is to simplify 1 - [(1 - x^2)/(1 + x^2)]^2 by making 1 = [(1 + x^2)/(1 + x^2)]^2, to get a common denominator. That is a common trick. I1 + x^2I makes no sense as 1 + x^2 is positive in any case. It's not wrong. It's just unnecessary.
 July 28th, 2013, 09:17 AM #3 Senior Member   Joined: Aug 2011 Posts: 149 Thanks: 0 Re: Can anyone tell what algebraic formula/trick is used her Thanks. I tried it: $1= $$\frac{1+x^2}{1-x^2}$$^2 \Rightarrow 1 = \frac{1+x^2}{1-x^2} \Rightarrow 0 = \frac{1+x^2}{1-x^2} -1 \Rightarrow 0 = \frac{1+x^2-(1-x^2)}{1-x^2} \Rightarrow 0 = \frac{1+x^2-1+x^2}{1-x^2} \Rightarrow 0 = \frac{2x^2}{1-x^2} \Rightarrow 0 = 2x^2 \Rightarrow 0 = x$ Is this right? And how do I use that 0?
 July 28th, 2013, 09:46 AM #4 Senior Member   Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116 Re: Can anyone tell what algebraic formula/trick is used her The general trick is: 1 - a/b = b/b - a/b = (b-a)/b Writing 1 as b/b, you get b as a common denominator. In your case, the demominator b = (1 + x^2)^2. You misread my original post.
 July 28th, 2013, 09:52 AM #5 Senior Member   Joined: Aug 2011 Posts: 149 Thanks: 0 Re: Can anyone tell what algebraic formula/trick is used her Edit: Thanks, got sqrt(2x^2) now, but where does that |1+x^2| come from? Do I just copy that common denominator to numerator and it's between | and | because it's under sqrt??
 July 29th, 2013, 04:50 AM #6 Senior Member   Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11 Re: Can anyone tell what algebraic formula/trick is used her The |1+x^2| has really no use because 1+x^2 is always positive anyway. you can leave it away if you want.
July 29th, 2013, 12:38 PM   #7
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Re: Can anyone tell what algebraic formula/trick is used her

Quote:
 Originally Posted by Pero The trick is to simplify 1 - [(1 - x^2)/(1 + x^2)]^2 by making 1 = [(1 + x^2)/(1 + x^2)]^2, to get a common denominator. That is a common trick. I1 + x^2I makes no sense as 1 + x^2 is positive in any case. It's not wrong. It's just unnecessary.
would say so too. why would someone leave the modulus there. since this is the case, (1+x^2) is exactly the same as |1+x^2|

it's probably another dumb text book question.

 July 30th, 2013, 01:05 AM #8 Senior Member   Joined: Aug 2011 Posts: 149 Thanks: 0 Re: Can anyone tell what algebraic formula/trick is used her My question was not that much about | and |. It was more abut how am I supposed to know that I have to write (1+x^2) in numerator.
 July 30th, 2013, 03:03 AM #9 Newbie   Joined: Jul 2013 From: Kathmandu, Nepal Posts: 1 Thanks: 0 Re: Can anyone tell what algebraic formula/trick is used her

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