If tan 1x tan 1y tan 1z pi 2
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If tan-1x+tan-1y+tan-1z = pi/2 prove that xy+yz+zx = 1
Answered by admin @
See put x = tanA , y = tanB and z = tanC ,
then A + B + C = pi/2 and
A+B = 90 - C
take tan both sides we get
tan (A+B) = tan (90-C) = cot C= 1/ tan C
or (tan A+ tan B)/(1- tan A tan B) = 1/tan C
cross multiply to get
(tan A + tan B) tan C = 1 - tan A tan B
or tan A tan C + tan B tan C = 1 - tan A tan B
or tan A tan C + tan B tan C + tan A tan B = 1
so xy + yz + zx = 1
Hence proved
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