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Question

If x1,x2,x3,x4, are roots of the equation x4x3sin2β+x2cos2βxcosβsinβ=0, then 4i=1tan1xi is equal to (where β(0,π2){π6})

A
πβ
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B
π2β
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C
π2β
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D
π22β
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Solution

The correct option is B π2β
If x1,x2,x3,x4 are roots of ...........................
We have S1=x1=sin2β
S2=x1x2=cos2β
S3=x1x2x3=cosβ
S4=x1x2x3x4=sinβ
so that 4i=1tan1x1=tan1S1S31S2+S4
=tan1sin2βcosβ1cos2βsinβ=tan1cosβ(2sinβ1)sinβ(2sinβ1)
=tan1cotβ=tan1(tan(π/2β))
π/2β

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