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If x + y + z = xyz, then prove that (2x)...

If x + y + z = xyz, then prove that `(2x)/(1- x^(2)) + (2y)/(1 - y^(2)) + (2z)/(1 - z^(2)) = (2x)/(1 - x^(2))cdot (2y)/(1 - y^(2))cdot (2z)/( 1- z^(2))`.

Text Solution

Verified by Experts

Let
`x=tan A,y=tan B,z=tan C`
Now `x+y+z=xyz`
`rArr tan A+tan B+tan C=tan A tan B tan C`
`rArr A+B+C=npi`
or `2A+2B+2C=2npi`
`rArr tan 2A+tan 2B+tan2C=tan2A tan 2B tan 2C`
`rArr (2tan A)/(1-tan^(2)A)+(2tan B)/(1-tan^(2)B)+(2 tan C)/(1-tan^(2)C)`
`=(2 tan A)/(1-tan^(2)A)(2tan B)/(1-tan^(2)B)+(2 tan C)/(1-tan^(2)C)`
`=(2tan A)/(1-tan^(2)A)(2tan B)/(1-tan^(2)B)(2 tan C)/(1-tan^(2)C)`
`rArr (2x)/(1-x^(2))+(2y)/(1-y^(2))+(2z)/(1-z^(2))=(2x)/(1-x^(2))(2z)/(1-z^(2))`
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