If x 1 , x 2 , x 3 , x 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 ∑ i = 1 4 tan − 1 x i is equal to

Question

If x 1 ​ , x 2 ​ , x 3 ​ , x 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 ∑ i = 1 4 ​ tan − 1 x i ​ is equal to

TestHub · Accepted Answer

i = 1 ∑ 4 ​ tan − 1 x i ​ ​ = tan − 1 ( 1 − ∑ x 1 ​ x 2 ​ + π x i ​ ∑ x 1 ​ − ∑ x 1 ​ x 2 ​ x 3 ​ ​ ) = tan − 1 ( 1 − cos 2 β − sin β sin 2 β − cos β ​ ) = ​ tan − 1 { sin β ( 2 sin β − 1 ) cos β ( 2 sin β − 1 ) ​ } = tan − 1 ( tan ( 2 π ​ − β ) ) = 2 π ​ − β ​

Question with Solution

Options:

Doubts & Discussion