If Tan−1x+Tan−1y+Tan−1z=π2 then
xy+yz+zx=1
x2+y2+z2+2xyz=1
x+y+z=xyz
∑x+∑yz=1+xyz
Let tan-1x=A,tan-1y=B and tan-1z=C. ⇒x=tanA,y=tanB and z=tanC. Given that tan-1x+tan-1y+tan-1z=π2 ⇒A+B+C=π2 ⇒tanAtanB+tanBtanC+tanCtanA=π2 ⇒xy+yz+zx=1