If `x_1, x_2, x_3,a n dx_4` are the roots of the equations `x^4-x^3sin2beta+x^2cos2beta-xcosbeta-sinbeta=0,` prove that `tan^(-1)x_1+tan^(-1)x_2+tan^(-1)x_3+tan^(-1)x_4=npi+(pi/2)-beta` , where `n` is an integer.

Explanation

Given equation is,

`x^4-x^3sin2beta+x^2cos2beta - xcosbeta - sinbeta = 0`

As `x_1,x_2,x_3 and x_4` are the roots of the above equation.

`:. sum x_1 = sin2beta`

`sumx_1x_2 = cos2beta`

`sumx_1x_2x_3 = cosbeta`

`x_1x_2x_3x_4 = -sinbeta`

Now, `tan(tan^-1x_1+tan^-1x_2+tan^-1x_3+tan^-1x_4) = (sumx_1-sum x_1x_2x_3)/(1-sumx_1x_2+x_1x_2x_3x_4)`

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