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Question
Solve the following for x :
`tan^(-1)((x-2)/(x-3))+tan^(-1)((x+2)/(x+3))=pi/4,|x|<1`
Solution
`tan^(-1)((x-2)/(x-3))+tan^(-1)((x+2)/(x+3))=pi/4`
`=>tan^(-1)((x-2)/(x-3))+tan^(-1)((x+2)/(x+3))=tan^(-1)1`
`=>tan^(-1)((x-2)/(x-3))=tan^(-1)1-tan^(-1)((x+2)/(x+3))`
`=>tan^(-1)((x-2)/(x-3))=tan^(-1)(1-(x+2)/(x+3))/(1+(x+2)/(x+3))`
`=>tan^(-1)((x-2)/(x-3))=tan^(-1)(x+3-x-2)/(x+3+x+2)`
`=>tan^(-1)((x-2)/(x-3))=tan^(-1)1/(2x+5)`
`=>(x-2)/(x-3)=1/(2x+5)`
`=>(x-2)(2x+5)=x-3`
`=>2x^2-4x+5x-10=x-3`
`=>2x^2=7`
`=>x=+-sqrt(7/2)`
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