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Question
Solve for x : tan-1 (x - 1) + tan-1x + tan-1 (x + 1) = tan-1 3x
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Solution
Given that tan-1(x-1)+tan-1x+tan-1(x-1)=tan-13x
⇒ tan-1(x-1)+tan-1(x+1)=tan-13x-tan-1x ...(1)
We know that, tan-1 A + tan-1 B = tan-1 `((A+B)/(1-AB))` and tan-1 A - tan-1 B = tan-1`((A-B)/(1+AB))`
Thus, tan-1(x-1)+tan-1(x+1)=tan-1 `((x-1+x+1)/(1-(x-1)(x+1)))`
`=tan^(-1)((2x)/(1-(x^2-1)))`
`=tan^(-1)((2x)/(2-x^2)) `
Similarly, `tan^(-1)3x-tan^(-1)x=tan^(-1)((3x-x)/(1+3x(x)))`
`=tan^(-1)((2x)/(1+3x^2)) `
From equations (1), (2) and (3), we have,
`tan^(-1)((2x)/(2-x^2))=tan^(-1)((2x)/(1+3x^2))`
`=>(2x)/(2-x^2)=(2x)/(1+3x^2)`
`=>1/(2-x^2)=1/(1+3x^2)`
`=>2-x^2=1+3x^2`
`=>4x^2=1`
`=>x^2=1/4`
`=>x=+-1/2`
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