If tan θ1, tan θ2, tan θ3, tan θ4, are the roots of the equation x4-x3 sin 2 β+x2 cos 2 β-x cos β- sin β=0 then tan (θ1+θ2+θ3+θ4) is

Question

If tan θ1, tan θ2, tan θ3, tan θ4, are the roots of the equation x4-x3 sin 2 β+x2 cos 2 β-x cos β- sin β=0 then tan (θ1+θ2+θ3+θ4) is (A)  sin β (B)  cos β (C)  tan β (D)  cot β

Tardigrade Admin · Accepted Answer

Given equation x4-x3 sin (2 β)+x2 cos (2 β)-x cos β- sin β=0 ∴ S1= sin (2 β), S2= cos 2 β, S3= cos β, S4=- sin β Now tan (θ1+θ2+θ3+θ4)=( s 1- s 3/1- s 2+ s 4) =( sin (2 β)- cos β/1- cos 2 β- sin β) =(2 sin β cos β- cos β/2 sin 2 β- sin β)= cot β

Q. If $tan θ_{1}, tan θ_{2}, tan θ_{3}, tan θ_{4}$ , are the roots of the equation $x^{4} - x^{3} sin 2 β + x^{2} cos 2 β - x cos β - sin β = 0$ then $tan (θ_{1} + θ_{2} + θ_{3} + θ_{4})$ is

$sin β$

$cos β$

$tan β$

$cot β$

Q. If tanθ1​,tanθ2​,tanθ3​,tanθ4​, are the roots of the equation x4−x3sin2β+x2cos2β−xcosβ−sinβ=0 then tan(θ1​+θ2​+θ3​+θ4​) is

sinβ

cosβ

tanβ

cotβ

Given equation x4−x3sin(2β)+x2cos(2β)−xcosβ−sinβ=0 ∴S1​=sin(2β),S2​=cos2β,S3​=cosβ,S4​=−sinβ Now tan(θ1​+θ2​+θ3​+θ4​)=1−s2​+s4​s1​−s3​​ =1−cos2β−sinβsin(2β)−cosβ​ =2sin2β−sinβ2sinβcosβ−cosβ​=cotβ

Q. If $tan θ_{1}, tan θ_{2}, tan θ_{3}, tan θ_{4}$ , are the roots of the equation $x^{4} - x^{3} sin 2 β + x^{2} cos 2 β - x cos β - sin β = 0$ then $tan (θ_{1} + θ_{2} + θ_{3} + θ_{4})$ is

$sin β$

$cos β$

$tan β$

$cot β$