If tanθ1,tanθ2,tanθ3,tanθ4 are the roots of the equation x4−x3sin2β+x2cos2β−xcosβ−sinβ=0 then tan(θ1+θ2+θ3+θ4)=
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)=S1−S31−S2+S4.
=sin2β−cosβ1−cos2β−sinβ=cosβ(2sinβ−1)sinβ(2sinβ−1)=cotβ.