Question

Solve and explain: tan2x= -cot(x+pi/3)

Answer 1

- cot (x + π/3) = - tan (π/2 - x - π/3)
                       = - (tan π/6 - tan x)/[ 1 + tanπ/6 tan x]
                       = (tan x - 1/√3) / [1 + tan x /√3]
tan 2x = 2 tan x /(1- tan²x) 

simplify and equate both sides
=> 2 tan x [√3 + tan x] = (1- tan²x) [√3 tan x - 1]
=> 2√3 tan x + 2 tan² x = √3 tan x - √3 tan³ x + tan²x -1
=> √3 tan³ x + tan² x + √3 tan x + 1 = 0

=> (√3 tan x + 1) (tan² x + 1) = 0
     
so    tan x = - 1/√3     => x = -π/6 or 5π/6  or 11π/6

Answer 2

tan2x=-cot(x+pi/3)=tan(pi/2+x+pi/3)=tan(5pi/6+x)

therefore

2x=npi+5pi/6+x,nEz

X=npi+5pi/6,nEz.