Number of real solution of the equation `(atn x+1) (tan x+3) (tan x+5) (tan x+7)=33`

Options

A. will be two in the interval `[-pi//2, pi//2]`
B. will be four in the interval `[-pi//2, pi//2]`
C. will be three in the interval `[-pi//2, pi]`
D. will be four in the interval `[-pi//2, pi]`

Answer

A, D

Explanation

`(tan^(2)x+8 tan x+15) (tan^(2) x+8 tan x+7)=33`
Let `tan^(2)x+8 tan x=p`, where `p ge -16`
`(p+15) (p+7)=33`
`rArr p^(2)+22p+72=0`
`rArr p=-18` (rejected) or `p=-4`
`rArr tan^(2) x+8 tan x+4=0`
so, `tan x+4= pm sqrt(12)`
`rArr tan x=-4+sqrt(12)` or `-4-sqrt(12)`