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Question

Solution set for the inequality 54sin2x+sin2xcos2x>cos2x is

A
2nπ+π3<x<2nπ+5π3,nZ
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B
nπ+π6<x<nπ+5π6,nZ
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C
nπ+π6<x<nπ+5π3,nZ
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D
nπ+π3<x<nπ+5π6,nZ
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Solution

The correct option is B nπ+π6<x<nπ+5π6,nZ
54sin2x+sin2xcos2x>cos2x5(1cos2x)+2sin22x>8cos2x5(1cos2x)+2(1cos22x)>8cos2x2cos22x+13cos2x7<02cos22x+14cos2xcos2x7<0(cos2x+7)(2cos2x1)<07<cos2x<12
but 1cos2x1
hence 1cos2x<12

Now from graph,


2nπ+π3<2x<2nπ+5π3
nπ+π6<x<nπ+5π6,nZ

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