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Question: If x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>, x<sub>4</sub> are the roots of the equation x<sup>4...

If x1, x2, x3, x4 are the roots of the equation

x4 – x3 sin 2b + x2 cos2b – xcosb – sin b = 0, then,

tan–1x1 + tan–1x2 + tan–1x3 + tan–1x4is equal to-

A

b

B

p/2 –b

C

p – b

D

– b

Answer

p/2 –b

Explanation

Solution

We have Sx1 = sin 2b, Sx1 x2 = cos 2b, Sx1 x2 x3 = cos b and

x1 x2 x3 x4 = – sin b

\ tan–1 x1 + tan–1 x2 + tan–1 x3 + tan–1 x4

= tan–1

= tan–1 sin⁡2β−cos⁡β1−cos⁡2β−sin⁡β\frac { \sin 2 \beta - \cos \beta } { 1 - \cos 2 \beta - \sin \beta } = tan–1 (2sin⁡β−1)cos⁡β(2sin⁡β−1)sin⁡β\frac { ( 2 \sin \beta - 1 ) \cos \beta } { ( 2 \sin \beta - 1 ) \sin \beta }

= tan–1 (cot b) = tan–1 tan (p/2 – b) = p/2 – b.