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This question today appeared in my maths olympiad paper:

If $\cos x + \cos y + \cos z = \sin x + \sin y + \sin z = 0$, then, prove that $\cos 2x + \cos 2y + \cos 2z = \sin 2x + \sin 2y + \sin 2z = 0$.

Can anyone please help me in finding out the solution of this equation ?

I have not gone any far in this solution.

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1 Answer 1

up vote 3 down vote accepted

Putting $a=\cos x+i\sin x$ etc,

we have $a+b+c=0$

and $a^{-1}=\frac1{\cos x+i\sin x}=\cos x-i\sin x$ $\implies a^{-1}+b^{-1}+c^{-1}=0\implies ab+bc+ca=0$

$a^2+b^2+c^2=(a+b+c)^2-2(ab+bc+ca)=0$

Now, $a^2=(\cos x+i\sin x)^2=\cos^2x-\sin^2x+i2\sin x\cos x=\cos2x+i\sin2x$ which is a special case of de Moivre's formula

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Brilliant answer ! Just one question, is there any other trigonometric proof or will this proof be accepted in IMO or any other such mathematical olympiad. –  Vishwesh Sep 1 '13 at 12:57
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@Vishwesh, have you understood the method? I think any logical method should be accepted in such competition –  lab bhattacharjee Sep 1 '13 at 13:02
    
Yes @lab bhattacharjee, I had the idea of de Moivre's theorem but could never think of applying it here. Fantastic method and flawless work. –  Vishwesh Sep 1 '13 at 13:05
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@Vishwesh, a related problem : math.stackexchange.com/questions/479726/… –  lab bhattacharjee Sep 1 '13 at 13:07
    
Ok, so if, nothing works, polar form can mostly come for rescue provided that solution must involve cos or sin. –  Vishwesh Sep 1 '13 at 13:09

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