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Question

If tanθ1,tanθ2,tanθ3,tanθ4 are the roots of the equation x4x3sin2β+x2cos2βxcosβsinβ=0 then tan(θ1+θ2+θ3+θ4)=

A
sinβ
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B
cosβ
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C
tanβ
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D
cotβ
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Solution

The correct option is B cotβ

From the given equation we get
S1=tanθ1+tanθ2+tanθ3+tanθ4=sin2β
S2=tanθ1tanθ2=cos2β
S3=tanθ1tanθ2tanθ3=cosβ
and S4=tanθ1tanθ2tanθ3tanθ4=sinβ
Now tan(θ1+θ2+θ3+θ4)=S1S31S2+S4.
=sin2βcosβ1cos2βsinβ=cosβ(2sinβ1)sinβ(2sinβ1)=cotβ.


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