If `A, B and C` are three values lying in `[0, 2pi]` for which `tan theta = K` then `tan""(A)/(3) tan ""(B)/(3) + tan ""(B)/(3)tan ""(C )/(3) + tan ""(C )/(3)tan ""(A)/(3)` is equal to ___________.

Answer

-3

Explanation

` tan theta =K `
`rArr = ( 3 tan ""(theta )/(3) - tan ^(3) ""( theta)/(3))/(1-3 tan ^(2)""( theta)/(3)) =K`
`rArr tan ^(3)""( theta )/(3) - 3K tan^(2)""( theta )/(3) -3 tan ""( theta)/(3) + K =0`
Roots of this equation are `tan ""(A)/(3), tan ""(B)/(3) , tan ""(C )/(3)`.
`therefore` Sum of roots taken two at a time = `sum tan ""(A)/(3) tan ""(B)/(3) = -3`.