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Question

If cosx=tany,cosy=tanz,cosz=tanx then

A
sin18
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B
12sin18
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C
2sin18
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D
3sin18
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Solution

The correct option is C 2sin18
We have cosx=tany

cos2x=tan2y=sec2y1 ......(1)

But cosy=tanz

secy=cotz

from (1)

cos2x=cot2z1

1+cos2x=cot2z=cos2zsin2z=cos2z1cos2z

But cosz=tanx

1+cos2x=tan2x1tan2x

1+1sin2x=sin2xcos2x1sin2xcos2x

2sin2x=sin2xcos2xsin2x

2sin2x=sin2xcos2xsin2x

2sin2x=sin2x12sin2x

(2sin2x)(12sin2x)=sin2x

2sin4x6sin2x+2=0

sin2x=3±942=3±52

But 3+52>1 whereas sin2x1

sin2x=352=6254=(51)24

sinx=512=2sin18

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