If x1, x2, x3, x4 are roots of the equation x4-x3 sin 2 β+ x2 cos 2 β-x cos β- sin β=0, then displaystyle∑i=14 tan -1 xi is equal to

Question

If x1, x2, x3, x4 are roots of the equation x4-x3 sin 2 β+ x2 cos 2 β-x cos β- sin β=0, then displaystyle∑i=14 tan -1 xi is equal to (A) β (B) (π/2)-β (C) π-β (D) -β

Tardigrade Admin · Accepted Answer

Sigma x 1= sin 2 β, Sigma x 1 x 2 = cos 2 β, Sigma x 1 x 2 x 3= cos β and x1 x2 x3 x4=- sin β ∴ tan -1 x1+ tan -1 x2+ tan -1 x3+ tan -1 x4 = tan -1((∑ x 1-∑ x 1 x 2 x 3/1-∑ x 1 x 2+ x 1 x 2 x 3 x 4)) = tan -1(( sin 2 β- cos β/1- cos 2 β- sin β)) = tan -1 ( cos β(2 sin β-1)/ sin β(2 sin β-1)) = tan -1 cot β = tan -1 tan ((π/2)-β) =(π/2)-β

Q. If $x_{1}, x_{2}, x_{3}, x_{4}$ are roots of the equation $x^{4} - x^{3} sin 2 β +$ $x^{2} cos 2 β - x cos β - sin β = 0$ , then $i = 1 \sum 4 tan^{- 1} x_{i}$ is equal to

$β$

$\frac{π}{2} - β$

$π - β$

$- β$

Q. If x1​,x2​,x3​,x4​ are roots of the equation x4−x3sin2β+ x2cos2β−xcosβ−sinβ=0, then i=1∑4​tan−1xi​ is equal to

β

2π​−β

π−β

−β

Q. If $x_{1}, x_{2}, x_{3}, x_{4}$ are roots of the equation $x^{4} - x^{3} sin 2 β +$ $x^{2} cos 2 β - x cos β - sin β = 0$ , then $i = 1 \sum 4 tan^{- 1} x_{i}$ is equal to

$β$

$\frac{π}{2} - β$

$π - β$

$- β$