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Question
Solve the following equation for x:
`tan^-1((2x)/(1-x^2))+cot^-1((1-x^2)/(2x))=(2pi)/3,x>0`
Solution
We know
`tan^-1x+tan^-1y=tan^-1((x+y)/(1-xy))`
`thereforetan^-1((2x)/(1-x^2))+cot^-1((1-x^2)/(2x))=(2pi)/3`
`=>tan^-1((2x)/(1-x^2))+tan^-1((2x)/(1-x^2))=(2pi)/3` `[becausecot^1x=tan^-1 1/x]`
`=>tan^-1((2x)/(1-x^2))=pi/3`
`=>2tan^-1x=pi/3` `[because2tan^-1xtan^-1((2x)/(1-x^2))]`
`=>tan^-1x=pi/6`
`=>x=tan pi/6`
`=>x=1/sqrt3`
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