If x+y+z=xyz then prove that x/(1-x^2)+y/(1-y^2)+z/(1-z^2)=xyz/{(1-x^2)(1-y^2)(1-z^2)}

Steps to prove x/(1-x^2)+y/(1-y^2)+z/(1-z^2)=xyz/{(1-x^2)(1-y^2)(1-z^2)}

Let us consider 

x=tanα, y=tanβ, z=tanγ

⇒x+y+z=xyz

⇒tanα+tanβ+tanγ=tanα.tanβ.tanγ

⇒tanα+tanβ=tanα.tanβ.tanγ-tanγ

⇒tanα+tanβ=tanγ(tanα.tanβ-1)

⇒(tanα+tanβ)/(1-tanα.tanβ)=-tanγ

⇒tan(α+β)=-tanγ

⇒tan(α+β)=tan(pi-γ)

⇒α+β=pi-γ

⇒2α+2β=2pi-2γ

⇒tan(2α+2β)=tan(2pi-2γ)

⇒(tan2α+tan2β)/(1-tan2α.tan2β)=-tan2γ

⇒tan2α+tan2β+tan2γ=tan2α.tan2β.tan2γ

⇒2x/(1-x^2)+2y/(1-y^2)+2z/(1-z^2)=2x/(1-x^2).2y/(1-y^2).2z/(1-z^2)

⇒x/(1-x^2)+y/(1-y^2)+z/(1-z^2)=xyz/{(1-x^2)(1-y^2)(1-z^2)}

Hence x/(1-x^2)+y/(1-y^2)+z/(1-z^2)=xyz/{(1-x^2)(1-y^2)(1-z^2)} is proved.

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