Question

If `x+y+z=xyz`, prove by trignometry that:
`(2x)/(1-x^(2))+(2y)/(1-y^(2))+(2z)/(1-z^(2))=(2x)/(1-x^(2)).(2y)/(1-y^(2)).(2z)/(1-z^(2))`

Answer 1

Let x=tanA, y=tanB, z=tanC
Now, `x+y+z=xyz`
`rArr x+y=-z+xyz`
`=-z(1-xy)`
`rArr (x+y)/(1-xy)=-z`
`rArr (tanA+tanB)/(1-tanAtanB)=-tanC`
`rArr tan(A+B)=tan(pi-C)`
`rArr 2A+2B=2pi-2C`
`rArr tan(2A+2B)=tan(2pi-2C)`