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if x 1 , x 2 , x 3 , x 4 are the roots of the equation x 4 -x 3 sin 2 beta + x 2 cos 2 beta – x cos beta – sin beta = 0 then ∑ i=1 to 4 tan -1 x i is ?
Let the roots be a,b,c,d...Then tan-¹ a + tan-¹ b + tan-¹ + tan-¹ d is equal to...tan-¹ [ (sum of roots - sum of roots taken 3 at a time) / (1 - sum of roots taken 2 at a time + product of roots) ]
Sum of roots taken 3 at a ime means abc + bcd + acd + abd ie all possible combinations of 3 roots out of 4.Sum of roots = sin 2ßSum of roots taken 2 at a time = cos 2ßSum of roots taken 3 at a time = cos ßProduct of all the roots = -sin ßPutting all the values:-> tan-¹[ ( sin 2ß - cos ß ) / ( 1 - cos 2ß - sin ß ) ]-> tan-¹[ ( 2sinßcosß - cosß ) / ( 2sin²ß - sinß ) ][2sinß - 1 cancels out.....]-> tan-¹[ cot ß ]-> tan-¹[ tan ( π/2 - ß ) ]ans:::::: π/2 - ß ......
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