Question

Find the general solution of the equation , 2 + tan x · cot x/2+ cot x · tan x/2= 0

Answer 1

let  (1 - tan² x/2) / 2  = Z

Tan x = 2 Tan x/2  / (1 - Tan² x/2) 
Cot x  = (1 - tan² x/2) / (2 Tan²x/2 )

given  2 + 1 / Z + Z  = 0    =>  Z² + 2 Z + 1 = 0    => Z = -1
     1 - Tan² x/2 = -2 
     Tan² x/2 = 3          =>  x/2 = n π + π/3

general solution:    x = 2 n π  + 2π/3         , n = integer

Answer 2

let  (1 - tan² x/2) / 2  = Z

Tan x = 2 Tan x/2  / (1 - Tan² x/2)  

Cot x  = (1 - tan² x/2) / (2 Tan²x/2 )

given  2 + 1 / Z + Z  = 0    =>  Z² + 2 Z + 1 = 0    => Z = -1

    1 - Tan² x/2 = -2  

    Tan² x/2 = 3          =>  x/2 = n π + π/3

general solution:    x = 2 n π  + 2π/3         , n = integer