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 ?

Question

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 ?

Faiz · Answer

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) ]

Faiz · Answer

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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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 ?

if x₁ , x₂, x₃ , x₄ are the roots of the equation x⁴ -x³ sin 2 beta + x² cos 2 beta – x cos beta – sin beta = 0 then ∑ i=1 to 4 tan^-1 x_i is ?

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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 ?

if x1 , x2, x3 , x4 are the roots of the equation x4 -x3 sin 2 beta + x2 cos 2 beta – x cos beta – sin beta = 0 then ∑ i=1 to 4 tan-1 xi is ?

2 Answers

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if x₁ , x₂, x₃ , x₄ are the roots of the equation x⁴ -x³ sin 2 beta + x² cos 2 beta – x cos beta – sin beta = 0 then ∑ i=1 to 4 tan^-1 x_i is ?