Simplify √((1+x)²+(1+1/x)²) - √(1+x²) - √(1+1/x²) ?
for all x > 0
Update:manjyomesando1 and Duke, the odd thing about this function is that it doesn't reduce for x < 0. So, technically speaking, manjyomesando1's proof needs a caveat. But Duke's sketch makes the derivation easy to see.
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√((1+x)²+(1+1/x)²) - √(1+x²) - √(1+1/x²)
= √[(1 + 1/x^2)(1 + x)^2] - √(1 + x^2) - √(1 + 1/x^2)
= (1 + x)√(1 + 1/x^2) - √(1 + x^2) - √(1 + 1/x^2)
= (1 + x - 1)√(1 + 1/x^2) - √(1 + x^2)
= x√[(x^2 + 1)/x^2] - √(1 + x^2)
= (x / x)√(x^2 + 1) - √(1 + x^2)
= √(x^2 + 1) - √(1 + x^2)
= 0
If You knew how much I like such questions, Scythian!
I must thank my Friend Buri, who has starred it, so I had a chance to read it and now can hardly resist the temptation to answer also, despite the fact You have already a correct answer by Manjyomesando1 above. Follow this link:
http://farm4.static.flickr.com/3246/3045454797_808...
I forgot to mention that the 3 right triangles on the picture are similar, but it's obvious of course.
(Added later) By the way, if x = Ï = (1 + â5)/2, then
|P₁P₃| = Ï (blue) and |P₁P₂| = 1/Ï (red).
Where on Earth do You dig such nice problems?