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Frustrating problem, Help! http://www.roadtoreality.info/viewtopic.php?f=22&t=102 
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Author:  ZWG [ 06 May 2008, 07:24 ] 
Post subject:  Frustrating problem, Help! 
Hi everyone. I Just started reading the book and working out solutions, I'll try and post some of my solutions next week. Chapters 4 + 5 reminded me of a particular problem from another book I have never been able to solve: Solve for x: Sin[Sin[Sin[Sin[x]]]]=Cos[Cos[Cos[Cos[x]]]]. Now the reason those chapters reminded me of it is becasue I can remember proving that if there did exist a solution it would not be real valued. If any one figures it out I would appreciate a hint prior to a full blown solution. Do I need to use the complex forms of the trig functions? Logs somewhere? Multiply by some tricky identity involving i? I have not thought about this problem in a while but I remember every approach I tried ended up as an algebraic nightmare. 
Author:  Sameed Zahoor [ 12 Jul 2008, 14:55 ] 
Post subject:  Re: Frustrating problem, Help! 
I am afraid this problem might not have a solution at all.Try proving that it does'nt have one.(Hint:use reductio adabsurdum!) 
Author:  dbcarm [ 04 Aug 2008, 22:46 ] 
Post subject:  Re: Frustrating problem, Help! 
Hi There It seems that this thing can't even get out of the gate. Sin(x) = Cos(x) iff x = PI*(1"4*j)/4 where j = 0 .. infinity. Now Sin(Sin(x)) = Cos(Cos(x)) iff Sin(x) = Cos(x) = PI/4 by the above which implies that Arcsin(PI/4) = Arccos(PI/4) but Arcsin(PI/4) = 0.903 radians and Arccos(PI/4) = 0.667 radians thus there is no x such that Sin(Sin(x)) = Cos(Cos(x)). 
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