We have,
#tan^-1sqrt((x(x+y+z))/(yz))+tan^-1sqrt((y(x+y+z))/(zx))+tan^-1
sqrt((z(x+y+z))/(xy))=pi#
It is clear that, #x > 0 , y > 0 , and z > 0#
We take Left Hand Side,
#LHS=tan^-1sqrt((x(x+y+z))/(yz))+tan^-1sqrt((y(x+y+z))/(zx)) #
#color(white)(.........................)+tan^-1sqrt((z(x+y+z))/(xy))#
#LHS=tan^-1sqrt((x^2(x+y+z))/(xyz))+tan^-1sqrt((y^2(x+y+z))/(
yzx)) #
#color(white)(.........................)+tan^-1sqrt((z^2(x+y+z))/(zxy))#
For simplicity we take ,
#color(brown)((x+y+z)/(xyz)=u^2 =>xyu^2=(x+y+z)/z=(x+y)/z+1...to(I)#
So,
#LHS=tan^-1sqrt(x^2u^2)+tan^-1sqrt(y^2u^2)+tan^-1sqrt(z^2u
^2)#
#LHS=tan^-1(xu)+tan^-1(yu)+tan^-1(zu)#
We know that,
#(II)color(red)(tan^-1X+tan^-1Y=pi+tan^-1((X+Y)/(1-XY)) , #
#when,XY > 1#
#X=xu and Y=yu=>X*Y=xuyu=color(brown)(xyu^2=(x+y)/z+1#
#i.e. X*Y >1#
Using #(II)# ,we get
#LHS=pi+tan^-1((xu+yu)/(1-xuyu))+tan^-1(zu) , tocolor(violet)(xuyu > 1)#
#LHS=pi+tan^-1((u(x+y))/(cancel1-(x+y)/z-cancel1))+tan^-1(zu)#
#LHS=pi+tan^-1(-(u(x+y))/((x+y)/z))+tan^-1(zu)#
#LHS=pi+tan^-1(-zu)+tan^-1(zu)#
#"Using "color(blue)( tan^-1(-X)=-tan^-1X# ,we get
#LHS=pi-tan^-1(zu)+tan^-1(zu)#
#LHS=pi#
#LHS=RHS#
