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| Just prove it (Posted on 2013-10-20) |
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Prove that:
If (x+sqrt{1+y^2})*(y+sqrt{1+x^2})=1,
then
x+y=0 .
Hyperbolic Approach
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Comment 4 of 4 | 
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Anything to avoid those square roots...
Let x = sinh(2X) and y = sinh(2Y), so that the equation becomes
[sinh(2X) + cosh(2Y)] [sinh(2Y) + cosh(2X)] = 1
cosh(2X)cosh(2Y)+sinh(2X)sinh(2Y)+sinh(2X)cosh(2X)+sinh(2Y)cosh(2Y)=1
cosh(2X + 2Y) + 0.5 sinh(4X) + 0.5 sinh(4Y) = 1
cosh2(X + Y) + sinh2(X + Y) + sinh(2X + 2Y)cosh(2X – 2Y) = 1
2 sinh2(X + Y) + 2 sinh(X + Y)cosh(X + Y)cosh(2X –2Y) = 0
sinh(X + Y) [sinh(X + Y) + cosh(X + Y)cosh(2X –2Y)] = 0 (1)
Since cosh(X + Y) >= 1 and cosh(2X – 2Y) >= 1 for all X, Y,
the expression in square brackets >= sinh(X + Y) + cosh(X + Y)
= exp(X + Y)
> 0
So the only solution from (1) is: sinh(X + Y) = 0 which gives X = -Y.
Since sinh is an odd function, it follows that sinh(2X) = -sinh(2Y) and
therefore that x + y = 0.
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Posted by Harry
on 2013-10-28 08:48:51 |
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