Question

Solve `(tan^(2) x+2sqrt(3) tan x+7) (cot^(2) y-2 sqrt(3) cot y+8) le 20` for x and y.

Answer 1

We have
`((tan x + sqrt(3))^(2)+4)((cot y-sqrt(3))^(2)+5) le 20`
Now, `L.H.S. ge 20`.
Therefore, we must have
`(tan x+sqrt(3))^(2)=0 and (cot y-sqrt(3))^(2)=0`
`rArr tan x=-sqrt(3) and cot y=sqrt(3)`
`rArr tan x=-sqrt(3) and tan y=1/sqrt(3)`
`rArr x=n pi-pi/3, n in Z` and `y=mpi+pi/6, m in Z`