If `A, B, C` are the angles of triangle `ABC` and `tan A, tan B, tan C` are the roots of the equation `x^(4)-3x^(3)+3x^(2)+2x+5=0`, if the fourth root of the equation is `tan D` then find `3tan D`.

Answer

5

Explanation

Let the fourth root of the equation be `tan D`
Now `A + B + C = pi`
Consider `tan (A + B + C + D)`
`= (sumtanA-sumtanAtanBtanC)/(1-sumtanAtanB+IItanA)`
`= (3 - (-2))/(1 - 3 + 5) = (5)/(3) rArr tan(pi + D) = tan D = (5)/(3)`