Solve the following equations:(i) tanθ+tan 2θ+tan 3θ=0(ii) tanθ+tan 2θ=tan 3θ(iii) tan 3θ+tan θ=2tan 2θ
(i) tanθ+tan 2θ+tan 3θ=0tan x+tan 2x+(tan x+tan 2x)1−tan x.tan 2x=0[tan x+tan 2x][1+11−tan x.tan 2x]=0tan x+tan 2x(2−tan x.tan2 x)=0tan x=tan(−2x)or tan x.tan 2x=2x=nπ−2x Or tan x.2tan x1−tan2x=2x=nπ Or2tan x1−tan2x=23x=nπ Or 2tan2 x=2−2 tan23x=nπ Or 4 tan2x=2x=nπ3Or tan2x=1/2x=nπ3Or x=mπ±tan−1(1√2),n,m∈Z(ii) tanθ+tan 2θ=tan 3θtan θ+tan 2θ=tan (θ+2θ)tan θ+tan 2θ−tanθ+tan2θ1−tanθ tan2θ=0[tan θ+tan 2θ][1−11−tanθ tan 2θ]=0[tan θ+tan 2θ][1−tanθ tan 2θ−11−tanθ tan 2θ]=0[tan θ+tan 2θ][−tanθ tan 2θ1−tanθ tan 2θ]=0tan θ=0 or tan 2θ=0 or tan θ+tan 2θ=0θ=nπ ornπ2or tanθ [1−tan2θ+21−tan2θ]=0θ=nπ or nπ2or tanθ=±√3θ=nπ or nπ3m,i,∈Z(iii) tan 3θ+tan θ=2tan 2θWe have,tan 2θ+tan θ=2tan 2θ⇒tan 3θ−tan 2θ=tan 2θ−tan θ⇒tan 3θ−tan 2θ=tan 2θ−tan θ⇒2 sin2θ sin 2θ=0⇒eithersin θ=0 Or sin 2θ=0⇒θ=nπ,n∈Z Or 2θ=mπ,m∈Z⇒ θ=nπ, n∈ Z Or θ=mπ2,m∈ Z