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

Question

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

Tardigrade Admin · Accepted Answer

We have, Sigma x1= sin 2β , Sigma x1x2= cos 2β Sigma x1x2x3= cos β and x1x2x3x4=- sin β ∴ tan -1x1+ tan -1x2+ tan -1x3+ tan -1x4 = tan -1( ( Sigma x1- Sigma x1x2x3/1- Sigma x1x2+x1x2x3x4) ) = tan -1( ( sin 2β - cos β /1- cos 2β - sin β ) ) = tan -1( ((2 sin β -1) cos β / 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 $tan^{- 1} x_{1} + tan^{- 1} x_{2} + tan^{- 1} x_{3} + tan^{- 1} x_{4}$ is equal to

$β$

$\frac{π}{2} - β$

$π - β$

$- β$

Q. If x1​,x2​,x3​,x4​ are roots of the equation x4−x3 sin2β+x2cos2β−xcosβ−sinβ=0, then tan−1x1​+tan−1x2​+tan−1x3​+tan−1x4​ 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 $tan^{- 1} x_{1} + tan^{- 1} x_{2} + tan^{- 1} x_{3} + tan^{- 1} x_{4}$ is equal to

$β$

$\frac{π}{2} - β$

$π - β$

$- β$