Find the general solution for cot^2θ-tan^2θ=4cot2θ

Steps to solve find the general solution for cot^2θ-tan^2θ=4cot2θ

⇒cot^2θ-tan^2θ=4cot2θ

⇒(1-tan^2θ)/tan^2θ=4/tan2θ

⇒(1-tan^2θ)(1+tan^2θ)=4tan^2θ(1-tan^2θ)/tan2θ

⇒tanθ(1-tan^2θ)(1+tan^2θ)-2tan^2θ(1-tan^2θ)=0

⇒tanθ(1-tan^2θ)(1+tan^2θ-2tanθ)=0

⇒tanθ(1-tan^2θ)(1-tan^2θ)^2=0

⇒tanθ(1-tan^2θ)^3(1+tanθ)=0

⇒tanθ=0, tanθ=1, tanθ=-1

⇒θ=nπ, θ=nπ+π/4, θ=nπ-π/4

Hence the general solutions of cot^2θ-tan^2θ=4cot2θ are θ=nπ, θ=nπ+π/4, θ=nπ-π/4

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