if tan-1x+tan-1y+tan-1z=π/2 show that xy+yz+zx=1?

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  tan-1x+tan-1y+tan-1z = π/2

=) tan-1x+tan-1y = π/2-tan-1z

=) tan(tan-1x+tan-1y) = tan(π/2-tan-1z)

= ) x+y/1-xy = cot(tan-1z)  [since tan(A+B)=tanA+tanB/1-tanAtanB & tan(π/2-A)=cotA]

=) x+y/1-xy = cot(cot-11/z)  [since tan-1A=cot-11/A]

= ) x+y/1-xy = 1/z

=) xz+yz=1-xy  [by cross multiplication]

=) xy+yz+zx=1

AS DESIRED //

 

 

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