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Question

If cosx =tany, cosy =tan z & cosz =tanx prove that sinx =siny =sinz

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Solution

We have : cos x =tan y.
cos2x=tan2y=sec2y1(1)
But cosy=tanzsecy=cotz
from (1),
cos2x=cot2z1
1+cos2x=cot2z=cos2zsin2z=cos2z(1cos2z)
But cos z =tan x
1+cos2x=tan2x(1tan2x)1+(1sin2x)=(sin2xcos2x)[1(sin2xcos2x)]2sin2x=sin2x(cos2xsin2x)2sin2x=sin2x(12sin2x)
(2sin2x)(12sin2x)=sin2x2sin4x6sin2x+2=0
by Quadratic Formula,
sinx=[3±94]2=(3±5)2
But (3+5)2>1whereassin2x1
sin2x=(35)2=(625)4=(51)24sinx=(51)2=2sin18
We can similarly show that
..sinx=siny=sinz=2sin18=(51)2

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