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Question

Solve the following equations:(i) tanθ+tan 2θ+tan 3θ=0(ii) tanθ+tan 2θ=tan 3θ(iii) tan 3θ+tan θ=2tan 2θ

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Solution

(i) tanθ+tan 2θ+tan 3θ=0tan x+tan 2x+(tan x+tan 2x)1tan x.tan 2x=0[tan x+tan 2x][1+11tan x.tan 2x]=0tan x+tan 2x(2tan x.tan2 x)=0tan x=tan(2x)or tan x.tan 2x=2x=nπ2x Or tan x.2tan x1tan2x=2x=nπ Or2tan x1tan2x=23x=nπ Or 2tan2 x=22 tan23x=nπ Or 4 tan2x=2x=nπ3Or tan2x=1/2x=nπ3Or x=mπ±tan1(12),n,mZ(ii) tanθ+tan 2θ=tan 3θtan θ+tan 2θ=tan (θ+2θ)tan θ+tan 2θtanθ+tan2θ1tanθ tan2θ=0[tan θ+tan 2θ][111tanθ tan 2θ]=0[tan θ+tan 2θ][1tanθ tan 2θ11tanθ tan 2θ]=0[tan θ+tan 2θ][tanθ tan 2θ1tanθ tan 2θ]=0tan θ=0 or tan 2θ=0 or tan θ+tan 2θ=0θ=nπ ornπ2or tanθ [1tan2θ+21tan2θ]=0θ=nπ or nπ2or tanθ=±3θ=nπ or nπ3m,i,Z(iii) tan 3θ+tan θ=2tan 2θWe have,tan 2θ+tan θ=2tan 2θtan 3θtan 2θ=tan 2θtan θtan 3θtan 2θ=tan 2θtan θ2 sin2θ sin 2θ=0eithersin θ=0 Or sin 2θ=0θ=nπ,nZ Or 2θ=mπ,mZ θ=nπ, n Z Or θ=mπ2,m Z


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