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प्रश्न
If `tan^(-1)((x-2)/(x-4)) +tan^(-1)((x+2)/(x+4))=pi/4` ,find the value of x
उत्तर
Given that
`tan^(-1)((x-2)/(x-4)) +tan^(-1)((x+2)/(x+4))=pi/4`
Taking LHS, we get:
`tan^(-1)((x-2)/(x-4)) +tan^(-1)((x+2)/(x+4))tan^(-1)((x-2)/(x-4)) +tan^(-1)((x+2)/(x+4))`
`=tan^(-1)[(((x-2)/(x-4)) +((x+2)/(x+4)))/(1-((x-2)/(x-4)) +((x+2)/(x+4)))]`
`=tan^(-1)([((x-2)(x+4)(x+2)(x-4))/(x^2-16-(x^2-4))])`
`=tan^(-1)[(x^2+2x-8+x^2-2x-8)/(12)]`
`=tan^(-1)[(2x^2-16)/(-12)]`
hence
`tan^(-1)[(2x^2-16)/(-12)]=pi/4`
`[(2x^2-16)/(-12)]=tan (pi/4)`
`=>(x^2-8)/(-6)=1`
`=>x^2-8=-6`
`=>x^2=2`
`=>x=+-2`
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