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Prove that tan theta+tan 2theta+tan 3theta=tan thetatan 2thetatan 3theta

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I think you have it wrong:

tan 3θ = tan(θ+2θ)
tan 3θ = (tanθ+tan2θ)/(1-tanθtan2θ)
tan3θ(1-tanθtan2θ) = tanθ+tan2θ
tan3θ - tanθtan2θtan3θ = tanθ+tan2θ
tan3θ - tan2θ - tanθ = tanθtan2θtan3θ

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